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What is the oldest work to have this trope?

This trope is prehistoric.

Greek mythology held that Eos, the goddess of the dawn, once asked Zeus to make her mortal lover Tithonus immortal, but Eos neglected to specify that he should remain eternally young as well. To Eos's regret, Tithonus grew older and older, shriveling away in his age; in some tellings, this caused him to eventually turn into a cicada.

(TVTropes catalogues this trope as Who Wants To Live Forever?. I drew the preceding example from there.)

(OK, let's not work ourselves into a place where everything is like everything else.)

Although TVTropes provides some examples of sucky immortality from ancient mythology, these are really just examples for a few unfortunate individuals, as a special punishment of the gods. Other mortals were made immortal and it was awesome for them.

But the idea of of immortality sucking in general, or just in modern fiction, probably dates back to Jonathan Swift's Gulliver's Travels. During Gulliver's visit to the island of Luggnagg, he encounters the Struldbruggs -- people whose bodies continue functioning forever, but also continue to age forever.

The Struldbruggs come from the general (mortal) population of Luggnagg. There are physical reasons why being a Struldbrugg sucks (one just keeps getting more and more decrepit) and social reasons (in order to keep the Struldbruggs from accumulating all the wealth and power, they are declared officially dead at 80 and forbidden from having wealth or owning property).

Hercules, a.k.a. Heracles, hit his master Chiron the centaur with a poisoned arrow as friendly fire during a battle. Chiron was immortal in the sense that he would never age and would live forever, and also immortal in the sense that he wouldn't die from the poison. Dude was the best healer ever, but he wasn't able to heal himself and the pain was unbearable. Zeus took pity of Chiron and took the centaur's life.

Since this is ancient greek mythology, this would have been contemporary to the story presented in duskwuff's answer.

@duskwuff gives one small example of the ancient Greeks discovering problems with immortality. However this is a relatively minor point in one fable.

More significantly, the ancient Greeks invented the concept of an afterlife for sinners as a place of eternal suffering, a concept most notably picked up by Christianity and called Hell. The Greeks called it Tartarus. The existence of Tartarus was not just a fable to the ancient Greeks (and the Romans who followed them and inherited their traditions), but was an established fact in their religion.

Note that this is distinct from the underworld of Hades, which is more akin to the Jewish concept of Sheol in that Hades takes everyone, both good and bad. Generally Tartarus is the destination for people who offend a specific god somehow (and sometimes for no fault of their own).

The Titans were the original inmates of Tartarus, after the "new gods" led by Zeus won the war against their Titan parents. Of those, perhaps the most notable example of eternal torture was Prometheus. Chained so he could not move, an eagle (or vulture depending on translation) would fly down every morning and disembowel him to eat his liver; and because of his immortality, the liver would have regrown by the next morning to allow the bird to do it all over again.

Mortals were then also sent to Tartarus for eternal torture too. Famous examples are Sisyphus, sentenced to drag a rock up a hill forever, or Tantalus, sentenced to eternal hunger and thirst with food and drink in sight but unreachable.

I'd say that pretty well qualifies as "immortality sucks"!

It seems to me that other answers describe intentional eternal suffering inflicted by gods instead of the immortality being itself the cause of boredom or unhappiness as asked in the original question. Still, as @duskwuff said, the trope is prehistoric. For the closest and earliest example I can think of, we need to look no further than at the earliest surviving great work of literature - Epic of Gilgamesh.

After losing his friend Enkidu, Gilgamesh was afraid of death. Siduri, the goddess of wisdom told him:

Gilgamesh, where are you hurrying to? You will never find that life for which you are looking. When the gods created man, they allotted to him death, but life they retained in their own keeping. As for you, Gilgamesh, fill your belly with good things; day and night, night and day, dance and be merry, feast and rejoice. Let your clothes be fresh, bathe yourself in water, cherish the little child that holds your hand, and make your wife happy in your embrace; for this too is the lot of man.

In his search for immortality, king Gilgamesh also meets Utnapishtim, his ancestor granted immortality for saving his family and animals from the great flood (predecessor of the Noah flood myth). He also dissuades him from chasing immortality, but eventually points him in the direction of the flower which can make him young again, but Gilgamesh loses it.

In the end, Gilgamesh does not attain immortality, he accepts his mortality and decides to "become immortal" by being a good ruler. One of the morals of the story is therefore to not chase immortality and instead live your mortal life to the fullest.

I would like to get some answers focusing on what the first mention of "healthy non-aging immortals" is, that grow tired of living.

I went looking for tales of eternal youth. There are a few kinds.

Sometimes the eternal youth is imposed by a lover. That would be the case of Endymion. Or also some variants of the tale of Ganymede.

Sometimes there is eternal youth as a result of an attempt to escape. That would be the case of Daphne (after being converted to a tree).

Depending of what version you get, the myth of the Moon Rabbit could fit in one of the categories above.

Some times the youth comes with a condition. For example, the tale of Kumbakarnan, who – if I am reading correctly – had to sleep for half of each year, and if awoke would die. He was awaken for war, and died in battle.

Another example, which comes closer to what we are looking for, is the condition of not leaving a particular land. Such would be the case of Odysseus and Calypso... or could be the case of Osin and the land of youth. These are a bit close to "grow tired of living", in that these characters left their eternal youth because of a desire to return home.

I am, of course, oversimplifying these myths.

Now, not exactly immortality... however, I think the tale of Yao Bikuni fits the bill. She ate the flesh of Ningyo. The Ningyo are human/fish yokai, eating their flesh grants eternal youth. Said yokai can also cause catastrophes, but those are other tales. Yao Bikuni got married, and saw his husband age and die, many times. She eventaully took her life.

From Wikipedia:

The story tells how a fisherman who lived in Wakasa Province once caught an unusual fish. In all his years fishing, he had never seen anything like it, so he invited his friends over to sample its meat.

One of the guests, however, peeked into the kitchen, noticed that the head of this fish had a human face, and warned the others not to eat it. So when the fisherman finished cooking and offered his guests the ningyo's grilled flesh, they secretly wrapped it in paper and hid it on their persons so that it could be discarded on the way home.

But one man, drunk on sake, forgot to throw the strange fish away. This man had a little daughter, who demanded a present when her father arrived home, and he carelessly gave her the fish. Coming to his senses, the father tried to stop her from eating it, fearing she would be poisoned, but he was too late and she finished it all. But as nothing particularly bad seemed to happen to the girl afterwards, the man did not worry about it for long.

Years passed, and the girl grew up and was married. But after that she did not age any more; she kept the same youthful appearance while her husband grew old and died. After many years of perpetual youth and being widowed again and again, the woman became a nun and wandered through various countries. Finally she returned to her hometown in Wakasa, where she ended her life at an age of 800 years.

According to the book "Death and the Afterlife in Japanese Buddhism", the tale of Yao Bikuni appears widely in the Tokugawa period, that would be 1603–1867, the book also claims there was evidence of a real Buddhist nun Yao Bikuni in the year 1449.

Addendum

Going to the Mahabharata, we find Ashwatthama, who was cursed:

(...) thou must have to bear the fruit of these thy sins. For years thou shalt wander over this earth, without a companion and without being able to talk with anyone. Alone and without anybody by thy side, thou shalt wander through diverse countries, O wretch, thou shalt have no place in the midst of men. The stench of pus and blood shall emanate from thee, and inaccessible forests and dreary moors shall be thy abode! Thou shalt wander over the Earth, O thou of sinful soul, with the weight of all diseases on thee.

-- Sauptika Parva, section 16 (source).

For what I have found, there are versions that say that the curse is for 3000 years (which seems to be most common, see), other say that the curse is until the end of the Kali Yuga. And some people claim that Ashwatthama is still alive (there are even reports of seeing him), see Aswathama Exists – Ashwathama Seen By People – Ashwathama Is Alive.

As per dating the Mahabharata, the usual dating is that it from the six century BC or older.

By the way, I would like to mention Revelations 9:6:

In those days men will seek death and will not find it; they will long to die, but death will escape them.

Petronius' Satyricon is famously quoted by the opening epithet of T.S. Eliot's poem "The Waste Land". The Cumaean Sybil, a prophetess, was blessed by Apollon to live as many years as there are grains in a handful of sand. When she declined his further advances, he did not give her resilience against aging. She finally shrunk to a size where she could be fit in a bottle where she yearned for death.

In classical literature immortality was either a curse or a blessing as seen from the examples in answers. Basically those born with it (like gods and various creatures) do well but if gifted it was a two-edged sword. The observation that when granted immortality you'll see everybody you know, love or hate eventually die is and was utterly obvious.

Don't overlook the medieval christian character of 'The wandering jew', doomed to walk the earth for eternity. Also the 15th century "La belle dame sans mercy" has a knight doomed to wander by (the personification) Tristesse, and taken away his capacity to feel by Mort; so feeling as a capacity of mortals was a clear concept then.

There are many interesting answers to this question already, but none of them seem to answer the question exactly.

As @Duskwuff's answer says, there have indeed been stories of regrettable immortality for a very, very long time.
However, as @Peter Pavlík points out, most if not all of these tend to be due to the type of immortality in question or a form of curse being tied to it, not the immortality itself.

TV Tropes

TV Tropes, as also referenced by @Duskwuff, lists the concept of undesired immortality as Who Wants to Live Forever?.

According to the examples from that page, the oldest example of ending an eternal existence being something to strive for probably comes from Buddhism.
Since there it is due to life inherently being a form of pain and the only immortality involved is through reincarnation and actually the natural state of all life, this does not exactly fit yet.

Literature

The oldest example from the "Literature" section appears to be The Tale of Norna Gest, estimated 1300 AD.

When he was recently born, his father had invited three seeresses, or norns, to foretell the child’s future. Two of the norns made good prophecies, but the last one was in a bad mood and when some rude guests enraged her, she cursed Gest to live no longer than the candle that burned beside his cradle. So the other norns extinguished the candle and told Gest’s parents to keep it, and Gest gained immortality — he cannot die before the candle is used up.

On the wish of King Olaf, Norna-Gest agrees to be baptized. After a time, King Olaf asks him how long he plans to live. Norna-Gest says that he wants to die, being three hundred years old. In the presence of King Olaf, he lays down on a bed and lights the candle. A priest gives him the last rites. When the candle burns out, he dies.

This story at least involves a human being attaining immortality though unnatural means deciding to end his life voluntarily.

This seems to be by far the oldest example of a story like this.

The next oldest story I could find from TV Tropes is The Ring of Thoth by Arthur Conan Doyle, first published in 1890.

Mr. John Vansittart Smith is a student of Egyptology who travels to France to see some papyri in the Louvre Museum. Tired from his visit, he sits in a corner of the museum and dozed. When he wakes up, the museum is closed and he is locked inside. Trying to get out he meets a strange man who seems to perform a ritual on a mummy. The curious man tells his story:
His name is Sosra and he was born 3500 years ago in Egypt. He was a priest of the god Osiris and discovered a remedy against death. He used it and shared it with his assistant, Parma. He wanted then to administer this elixir to his fiancee, Atma, but she died just before. He went desperate because he couldn't join her in the afterlife. So he wanted to become mortal again. Parma reported him that he had found an antidote but he had decided to use it for himself so he can die and meet Atma beyond. Sosra realized that the antidote was in the ring of Thott but Parma had hidden it and he took his secret to the grave. 3500 years later the mummy of Atma is found by French archaeologists and repatriated to the Louvre. Sosra succeed to be hired in the museum, found the Atma sarcophagus and got the ring containing the precious elixir.
At the end of his incredible story Sosra accompanies Vansittart to the exit. Two days later he learns in the newspapers that a man was found dead in the museum entwined in the arms of a mummy.

Most if not all other examples from literature seem to be newer than at least the 1920s.

Since death and the desire to avoid it are, and have always been, a fairly important part of human existence, the actual origin of this trope may be lost to time or not exist at all.

This question asks about the origin and the oldest example (two questions). It also asks specifically for tales where immortality is itself the curse, not some torment or suffering for eternity such as Tithanos or Struldbrugss. I will try to answer both with best known dates:

The trope of unaged immortality is a simple extension of moral lessons against gluttony by fulfilling greedy desires in excess. Essentially, "If you like X so much, have all you want and see what happens!"

The first known literary works telling of a mortal actually cursed with eternal health and undying are many variations of The Wild Huntsman from Rhineland folklore. The most famous of these is Sir Walter Scott's The Wilde Huntsman published in 1796, which itself is an imitation of the German folk tale by Gottfried August Bürger.

"Oppressor of creation fair!

Apostate spirits’ hardened tool!
Scorner of God! Scourge of the poor!

The measure of thy cup is full.

"Be chased for ever through the wood,

For ever roam the affrighted
wild; And let thy fate instruct the proud,

God's meanest creature is His child

• Sir Walter Scott

Many derivative works of the Huntsman immortality curse exist, most notably Heinrich Heine's Flying Dutchman published in his 1833 satirical memoire entitled Aus den Memoiren des Herrn von Schnabelewopski. Heine's work in turn inspired the opera of the same name by Wilhelm Richard Wagner composed in 1843.

The many various wild huntsman stories themselves date back before written records and have always had an undying huntsman, originally known to be Herne The Hunter (a ghost), the old High German Wuotan, or the Norse god Odin. These have passed through oral folklore since ancient times, however they have always cast the Huntsman as a spirit or a god.

Christian influences seem to have adapted the story turning the Huntsman into Bürger's idea of an obsessed civic annoyance cursed with undying for gleefully murdering wildlife, trampling crops under hoof, raising a ruckus, and defying God's own command. The inspiration for Bürger's adaptation has not been recorded however there is precedent which precedes Judaism.

The origin was very possibly the well-known story of Gilgamesh's obsessive and ill-advised quest for immortality. The theme of that story is that he was fortunate not to achieve it, so a simply literary extension allows a good storyteller to simply grant the greedy Gilgamesh his wishes and account the consequences. While many versions of this story have been found, a history and synopsis of one unfinished version which could have ended with an immortal and young Gilgamesh follows:

The Babylonian Sippar tablet is one of the oldest works of written literature, dating to between 888-850 BC during the reign of Nabu-apla-iddina. It is one of many tablets recovered documenting Gilgamesh who was obsessed with achieving immortality out of fear from the death of his friend Enkidu. This tablet in particular tells of his dismay over several failed attempts. He then consults the god Shamash who quarrels and tells him the quest is futile, and the alewife Siduri who urges Gilgamesh to be content with the simple pleasures of life. Shamash does feel pity and tells Gilgamesh he should pursue youth instead of immortality, giving instructions on how to find the rejuvenating box-thorn plant. Gilgamesh found the plant but it was stolen before he could test it.

While this tablet is incomplete, the message is clearly warning against the futility and uselessness of immortality and the quest for it.

Likewise, in another tablet Gilgamesh finally reaches Utnapishtim, the only immortal, to ask how he achieved it. The response is similar:

"There is no permanence. Do we build a house to stand forever, do we
seal a contract to hold for all time?…. When the Anunnaki, the judges,
come together, and Mammetun the mother of destinies, together they
decree the fates of men. Life and death they allot but the day of
death they do not disclose."

It is not known how or if this story reached 18th century German authors however it serves as an origin for the impropriety of immortality and eternal youth. Unfinished legends and myths are strong sources for writer inspiration. It seems reasonable that at some time a writer chose to grant Gilgamesh his wish and account the consequences. "What if Gilgamesh found immortality?" The narrative is clear that his eternal life would logically follow unfavorable paths.

Happyaku Bikuni (After 552 AD) A Japanese folk tale about a buddhist nun who was inadvertently fed meat from a mermaid gained eternal youth and was saddened as her family eventually all died. This has been carried along orally with no known origin. Buddhism reached Japan in 552 AD, the tale is at most that old. The mythical Ningyo (mermaid) itself which she ate is traditionally said to be a goddess who if you capture her and bite her flesh you will have eternal youth and beauty. Other ageless unwritten stories about women who have succeeded in this pursuit could also have inspired this trope in visitors to Asia.

Other Notable Immortals in Literature

• Appolonius of Rhodes wrote a poem about Endymion (c. 250 BC) who was blessed with eternal youth by Zeus however he was also in eternal sleep.




• Jonathan Swift's Gulliver's Travels (1726). Gulliver's finds the island of Luggnagg, where the Struldbruggs -- bodies continue functioning forever, but also age.




• Nornagestr (Norna Gest) - 14th century - Could not die before a candle was burnt out, yet he still became "stricken with years." Converted to Christianity, lit the candle, and died.