Polgara, and to a lesser extent Belgarath and the other sorcerers, have shades of this in the Belgariad and Malloreon. Polgara gets to raise, live alongside and bury an entire succession of hidden royalty, as well as a lover and many many friends, while Belgarath has been mourning his dead wife for 5,000 years. At one point it is mentioned that the serenity of the Vale of Aldur and the continuity of the World Tree within it is all that keeps them sane.
Jason and Anna in Finding Gaia get weary of their extended lives, and have different ways of coping with it.
This defines the Nonmen in R.Scott Bakker's Second Apocalypse series. They gained immortality from the Inchoroi, but because their minds are only equipped to deal with a few centuries worth of memories, the millennia have driven almost all of them insane, some to the point where they commit unforgettable atrocities just so they'll have something to remember. The fact that their immortality came with a Gendercide also makes their unending lives utterly pointless.
Life, the Universe and Everything: Wowbagger the Infinitely Prolonged, having seen and done everything there is to see and do, decides to dedicate the rest of his existence to insulting every single living being in the universe — in alphabetical order. It is interesting to note that the Guide points out that those who are naturally immortal are born with the psychological capacity to cope with immortality and would not suffer from this trope; Wowbagger's immortality was thrust upon him by accident, which is why he has such a hard time of it.
Marvin has lived several times longer than the lifetime of the universe through various Time Travel mishaps. He hates it. He gets to die eventually.
Dirk Gently's Holistic Detective Agency: Professor Urban Chronotis, is, for no reason the book makes clear, an apparent immortal who is so old that he's forgotten most of his origins. He fears that his eventual fate will be to "sit alone in a darkened room, sans teeth, sans eyes, sans taste, sans everything but a little grey old head..." (Knowing the back story clears this up significantly: the story was originally going to be a Doctor Who story, with Reg as a fugitive Time Lord.)
In Cordwainer Smith's Instrumentality stories, the Norstrilians have unlimited access to stroon and are potentially immortal. But stroon has "queer side-effects, so that most Norstrilians preferred to die in a thousand years or so."
Mars: "Life's only precious because it ends, kid. Take it from a god. You mortals don't know how lucky you are."
Lloyd Alexander's short story, "The Stone", was about a man who found a stone that made him live forever - by stopping change, making everything on his farm exactly the same, day after day after day. He couldn't get rid of it easily, either - the stone was a Clingy MacGuffin.
The protagonist from Isaac Asimov's Bicentennial Man is a robot, who can quite literally live forever by repairing himself whenever necessary. However, his wish is to be human. In an age of cybernetic prosthetics and replacement body parts, the boundary between man and machine blurs - and eventually, he gets himself legally declared human, but only after he introduces inevitable decay into his robotic brain, ensuring that he will eventually die like a human, rather than live forever as a robot.
Also from Isaac Asimov, The Last Answer has God harnessing dead beings' inevitable desire to snuff it, in an effort to figure out how he himself can do the same.
Natalie Babbitt's Tuck Everlasting. The main character sacrifices the chance to live forever with an immortal who loves her for a normal life. Note usually this kind of character has to choose between eternal life and a mortal love; here, she can get immortalityand love... but gives up both. The Tucks are all, some more than others, unhappy about having to watch the world change around them and people they know (and sometimes love) pass away.
In the Culture novels by Iain M. Banks, citizens of the Culture have the option of dying of old age (after several centuries of life), or having their age stabilized to become effectively immortal (assuming accidents don't happen). However, there's a cultural bias towards dying when your time is up, and choosing immortality is thought of as immature - although the Culture is all about IDIC and this bias is probably not a constant. Multiple other options are also present; you can Ascend to a Higher Plane of Existence or store yourself to be revived at a later date (either physically or electronically). These options can be combined; not uncommonly, those who elect to die also upload a version of themselves (presumably with tweaks so it won't get tired of living) so that their memories and experiences are not lost. The machine citizens, and especially the Minds, are immortal by default (again, barring no accidents), but emotional trauma can very rarely lead to a machine mind committing suicide.
In Stephen Baxter's Manifold: Space, Nemoto keeps herself alive with advanced medical treatments for well over a thousand years, so she can deal with the problem of the alien Gaijin (and whoever the Gaijin are fighting). She doesn't seem to enjoy it much, and becomes extremely crotchety — but she's too much of a control freak to leave things in anyone else's hands.
In Baxter's Xeelee Sequence of novels, a group of people known as Jasofts gain immortality. However they suffer in that ultimately, they can only hold one thousand years' worth of experiences, and live many times that, sometimes able to vividly remember events, before seeing something which brings back other memories and pushes those away.
Schmendrick, whose mentor made him immortal until he could come into his real power. When his power transforms the unicorn into a human woman, he tries to tell her about the beauty of things that can die, a lesson she learns all too well before she regains her immortality (and he loses his).
Amalthea wishes to choose death rather than become immortal and fall out of love with Lir.
In the Morgaine Cycle the gates can be programmed to provide a kind of immortality for their users. The user's physical condition can be scanned upon entering a gate for the first time, and subsequently each time they enter a gate they will emerge on the other side in that original condition, thus resetting their biological clock to its earlier state. This pisses Vanye the hell off when he learns he's been thus fixed a few years before his physical prime.
In the Fortress Series, we have Tristen, who is a wizard's shaping and was around for centuries as Barraketh, as long as history knows anyway. His memories in Fortress of Dragons imply millennia. It is also implied that his previous life never died exactly; there are no graves for any of the original five Sihhe.
All Men Are Mortal, by Simone de Beauvoir. Besides the usual miseries of the immortal, Fosca is tormented by unreliability of people. He chose to become immortal so that he could make a political difference, only to find out that it's not time you need, it's people.
Sir Arthur Conan Doyle's story "The Ring of Thoth" is about an Egyptian who discovers, and injects himself with, an elixir which grants near-immortality, only to have the one he loves die of the plague before he can give her the elixir. The story is about his four-thousand-year search for the only poison strong enough to overpower the elixir.
Lyndon Hardy's Master of the Five Magics series. In Riddle of the Seven Realms, it is revealed that demons adopt various hobbies to avoid succumbing to this trope. Palodad the Reckoner, under its influence, turns into a ChessmasterOmnicidal Maniac.
In Brian Jacques's Castaways of the Flying Dutchman series, the main characters are an immortal boy and dog. Leaving aside the fact that the boy is stuck at age 14 forever, they have to leave everyone they ever get close to before someone notices that they don't age.
Mercedes Lackey's auto-racing elves who get involved with humans are in fact traumatized by the deaths of the people around them, especially lovers and spouses, but they live with it.
One of the stories in Ursula K. Le Guin's Changing Planes discusses a plane where, it is rumored, immortals live. As it happens, there are a handful of them, the result of bites by a certain fly. They don't get eternal youth, and are condemned to endless agony. One of the plane's natives, who watches over one immortal in particular, notes that eventually the people bury their immortals, and over centuries their suffering apparently condenses them into a diamond. The narrator asks if the native is afraid of the flies because of this, and is told "There's only one"; as there are many flies on the plane, the narrator theorizes that there is one immortal fly that curses the bitten with immortality.
A tree's fruit comes with the warning that it brings eternal life and despair. The White Witch eats it, and from her expression, the title nephew understands the warning. Narnia is then protected from her by a tree grown from one of the apples; she cannot stand to come near it afterward.
The Witch tries to tempt Digory (the Nephew) into eating the apples and living forever as well, invoking We Can Rule Together. Digory promptly responds that he'd rather live a normal amount of time and go to heaven rather than stay living and watch all his friends die.
In Lords and Ladies, the not-nice kind of The Fair Folk invade Lancre. Granny, who seems to be going senile (turns out she's not), is mocked by the ever-young elf Queen for growing old... and Granny turns the insult right around:
"What don't die can't live. What can't live don't change. The smallest creature that dies in the grass knows more than you ... you've lived longer'n me but I'm older'n you, and I'm better'n you."
Eric: Eric's third wish is to live forever. The demon granting his wishes sends Eric back in time to the beginning of the universe. He's not too thrilled with the prospect of having to spend several billion years as the only living thing in it.
Carrot uses the trope name in an inspirational speech in one of the Watch books. Sergeant Colon snarkily replies that he doesn't know, ask him again in a few centuries.
In A Hat Full of Sky, the hiver. It has existed longer than existence itself, it is close to being omniscient and thus experiences every smell, sight, etc. along with having total recall. Simply put: it experiences everything there is and has been all at once, and this drives it to suicide. Death is the only thing it does not know though, so it possesses mortal creatures in an attempt to understand death and how to actually die.
In Mort, Mort quickly realises that Death wants to lose their Duel to the Death, and also that he really doesn't want the Klingon Promotion that comes with winning (which doesn't stop either of them fighting to the best of their ability, until Death figures out the third option). In Soul Music Death explains that he couldn't extend Mort and Ysabell's lives, because granting them immortality wouldn't have been the same thing and they didn't want it. Albert, on the other hand, reckons an eternity of not-quite-life as Death's manservant suits him just fine, given what's waiting for him on the other side.
Strata: People working for "The Company" can get treatments to which make them effectively immortal. Despite this, most people don't live more than a few hundred years, because they grow tired of life where they have already done everything they can do. Not that they commit suicide as such; they just keep doing more and more dangerous things to get the same excitement, and eventually one of them goes wrong.
A particularly disturbing twist on this trope is Claudia from Anne Rice's The Vampire Chronicles, a young girl who is made into a (theoretically immortal) vampire and matures mentally and emotionally, but not physically. This leaves her perpetually dependent on others, embittered and perhaps not entirely sane. It is later revealed that vampire law prohibits the making of child vampires for precisely this reason. Several characters decide to commit suicide because they are bored with eternal life or just tired of living in "The Savage Garden."
Anita Blake novels: One of the more disturbing vampires, called Valentina, was turned at the age of eight by a vampire pedophile who was bringing over children to be his permanent companions. The few vampires turned as children who survive a few centuries and described as twisted things. In Valentina's case, "(She) was taken before her body grew large enough for much physical pleasure. She has turned such energies into other avenues of interest," which in this case means torturing others.
In Jeffrey Sackett's Mark of the Werewolf, the main character is cursed with immortality. This results in him forgetting anything beyond two hundred years past (including his own name), transforming into an Ax-Crazy werewolf every night of the full moon (unless outfitted with a Restraining Bolt beforehand), and being invulnerable to damage in any form. The book revolves around his attempts to figure out how to die.
In The Book of Mormon, the Three Nephites are granted the gift/curse to live until Christ returns, and were still going strong roughly 400 years later.
Unusual use of an elf with this trope: Drizzt Do'Urden from R.A. Salvatore's Forgotten Realms books. He spends a number of books angsting over eventually losing his friends and loved ones and even debating whether or not to get into a relationship because of it. He gets over it by adopting a carpe diem mentality.
In Flash Forward by Robert J. Sawyer, one of the main characters is approached by a man offering to increase his lifespan through new medical techniques. In the second "Flash," he had seen that if he accepts the offer he has the chance to live forever. He saw the future of mankind, with humans eventually dismantling the Earth and using the materials to build a Dyson Sphere, and then spreading throughout the galaxy. He saw himself on another world, in a new, mechanical body. The offer is made later, but we never learn whether he accepted it.
The character Tithonus from Greek Mythology and the Struldbrugs from Jonathan Swift's novel Gulliver's Travels both suffer the torment of eternal life without the benefit of eternal youth. Although they never die, they age at the normal rate, and so are condemned to an eternity in decrepit ancient bodies.
In the third part of Gulliver's Travels, there are people in Luggnagg who live forever. However, they still age normally and suffer greatly from senility. The Luggnaggians thus do not particularly desire immortality.
Death isn't called 'Eru's gift' for nothing. It is heavily suggested that fear of death is actually created by Morgoth to mess with humans. It's implied that for Men there is a possible life after death, outside Arda; while Elves, who are bound to Arda, may not survive when it is undone. The elves are bound to the world forever; even if they die, they are stuck in the halls of Mandos and can eventually become re-embodied in Valinor. Thus, they experience the entire lifetime of the world, and time weighs on them. The Silmarillion says that even the Valar will eventually envy humans' ability to die and leave the world.
Naturally, immortality ceases to be all it's cracked up to be for elves who fall in love with mortal men, and are forced to contemplate a literal eternity of grieving for them. Which is perhaps why, on two occasions, the elf was granted the power to die, and to follow their beloved out of the world.
In The Lord of the Rings, at one point Sam, when pondering Smeagol/Gollum (who is a Hobbit-like creature given, if not immortality, a lifetime extended centuries beyond his proper span), gets a little hint of what Gollum's existence has been like. Preserved out of his time by the Ring, far beyond kin, friends, and everything he knew, a stranger in a different world. Utterly alone, living endless days in the dark, entrapped by a Ring he can't escape and can't give up and loathes. Bilbo himself, in spite of his happy life, begins to feel the weight of his increased lifespan toward the end. When he surrenders the One Ring, he feels an instant sense of relief as his natural state of being returns.
This is the entire premise behind the Tide Lords series.
In Gaunt's Ghosts, the Tanith First and Only have a battlecry: "Men of Tanith! Do you want to live forever?"
Golden Boy stopped aging at his early twenties and shows no signs of aging. Since he's also invulnerable as part of his Combo Platter Powers, it's unlikely anything else will kill him either. His situation is somewhat aggravated by the fact that he's already cut off from his peers, who despise him for rolling over and testifying at McCarthy's anti-Ace hearings. Thus he is presented with the possibility of an immortality of being reviled and hated by anyone who knows who he is.
There's also Demise, who had already died from the virus and had been resurrected by Tachyon and may well have been able to live forever had his corpse not been reduced to ashes after he'd been killed for the nth time.
The Sleeper wakes up young and healthy (relatively) every time he sleeps, ever since the first outbreak in 1946. He lampshades this with the occasional "they sure didn't (X) like that when I was your age."
The Misenchanted Sword by Lawrence Watt-Evans. The main character receives a sword which will not allow him to die until he has killed 100 men with it. He wisely decides to live forever and not kill people, but reverses this decision because of the other problems with the sword. It doesn't save him from age, and it doesn't protect him from injury. At one point he nearly "dies" because of blood loss but is still mysteriously alive the next day. When he discovers his eyesight is fading, he goes off to kill 100 men and rid himself of the sword before he becomes unable to do so. The end of the book reverses the trope once more: The main character discovers magic that will keep him young and can be added onto the immortality the sword already gives him. Once he does so, he's quite happy to be immortal.
This becomes a vague plot point in Brisingr, when Eragon and Roran discuss Eragon's immortality. Eragon has concluded that this forces him to marry an elf, who are all immortal, rather than a human woman, and so thus his drooling over Arya (who refused him multiple times) is justified.
Some fantasy fiction, such as R.A. Salvatore's Forgotten Realms novels and the Fighting Fantasy gamebooks published in the 1980s and 1990s, depicts demons as being horribly bored and depressed by their endless existence in Hell, the novelty of torturing their servants and fellow demons having long since worn off. Of course, their boredom and frustration make them all the more eager to torture humans and other mortal creatures when they find their way to our world.
At first played straight with the Cullens (and maybe other vampires). Later averted for Edward, since Bella becomes an immortal vampire too and they have a child together that brings purpose to Edward's life.
Rosalie is especially bitter about being immortal, believing that a "happy" ending would be for them all to be dead and buried long ago.
Aro of the Volturi is somewhat of an example, according to Word of God: "After three thousand years, you start to go nuts. That is what was intended for Aro in the book, and I hope it's in the movie."
Byron's closet drama Manfred is a melodramatic refashioning of the Faust legend. When he summons seven Spirits who swear to do his bidding, he asks not for power but for forgetfulness. The entire play is his search for death, since the star under which he was born cursed him to live forever.
The "naturally immortal species are immune to this" angle is subverted with the Tiste Andii, who have become rather bored and depressed with life since they have nothing to do and nothing to look forward to. Their Warrior Prince Anomander Rake tries to overcome this by involving his people in conflicts they don't actually have anything to do with just to give them a reason to fight and to go on.
Nathan Brazil, the immortal Guardian of the Well of Souls from Jack Chalker's Well World saga, suffers this from time to time because his role as the emergency repair man for the universe means he absolutely cannot die. The universe simply won't allow it. Every time he reboots the universe (it's happened at least five times so far) he's been forced re-live all of human history until the next time he's needed. Oh, and did I mention that rebooting the universe requires him to kill every living creature in creation? He's tried various coping strategies, from blanking his memories to recruiting another to be his immortal companion (they had a falling out after 15,000 years or so) to "accidentally"recreating himself as a woman during the latest reboot, probably in the hope that will make it all different this go-round.
While the denizens of John Varley's Eight Worlds series may all be potentially immortal (due to really advanced medical technology), very few of them actually live much beyond 300 years, largely due to the effects of this trope.
Observation on the Spot: As Ijon Tichy finds, most people who've tried immortality in a seemingly "everything-is-possible" society of Lusania, didn't really like it. It seems that mortals' psychology (the guys in question are aliens, but surprisingly humanlike psychologically) is simply ill-suited to immortality. There are just six immortals who finally learned to cope and hasn't ended it all in different ingenious ways, and all of them don't like to talk about it.
Ijon Tichy meets with an inventor who created an immortal soul. However, for that, the body has to be destroyed, and the soul is kept in a box, without any external stimuli. Tichy realizes that this is a fate worse than death. He tells to the inventor that people don't want immortality; they just want to live.
In "The Twenty-first Voyage" from The Star Diaries, all of the people turned immortal by advanced technique were driven to suicide by the immense machinery that encircled them all the time. As to the example from Observations on the Spot the rub lies in the fact that the nanomachines, as they gradually replace the living tissues of the immortalized, cause biological cells to eventually die out, making the subject became something that's no longer a living organism.
Almost the same story happened in Strugatsky BrothersNoon Universe (specifically, Far Rainbow), except this time immortality was brought not by nanomachines, as in Lem's case, but by full-body cyborgization (although, given the state of technology in Noonverse at the point, nanomachines still would play the role, at the very least). Only one among the subjects remained stable and sane in the end, and it's implied that it was only because he wanted to observe the society and snarkat its failings.
Pandora's Star/Judas Unchained and the later Void Trilogy has a range of attitudes to immortality - everything from I-can't-take-it-any-more boredom to Ascend to a Higher Plane of Existence (but keep your body so you can visit the material world) to "I will stay in the real world for a thousand years just to make a difference." It's implied that the race as a whole has been modifying itself to cope, though there are few true Methuselahs in the real world.
Diana Wynne Jones's The Homeward Bounders, which features multiple characters (including a number of mythological figures) condemned to eternal existence bouncing from world to world by the whim of malevolent beings known only as Them. In the end, They are defeated, but the protagonist elects to keep wandering in order to keep Them away because he cannot accept the place his world has become in the intervening time as "home."
In Mockingbird, by Walter Tevis of The Man Who Fell to Earth fame, the robot Spofforth — doomed to live forever — constantly fantasizes about committing suicide, but his programming prevents him from attempting it until the last page, when the protagonist helpfully pushes Spofforth off the edge of a tall building.
In The League of Peoples Verse, Oar's race is unable to die. Their minds eventually become exhausted and shut down, but their bodies continue to live on and on. As a result, they view death as something sacred.
The short story "Mortal Gods" (collected in Maps in a Mirror) has an alien race venerating mortal humans because they all, eventually, die — something the aliens are unable to do.
Card addressed this again in The Worthing Saga, which has a particularly sub-par method of "immortality"—go into a comatose state for an indefinite amount of time, and you won't be any older when you come out of it. Just about everyone who can use this does so, but outsiders tend to realize this sort of extended life doesn't allow for any more time spent doing things, and does result in your poorer friends dying significantly before you. (Also, it messes up their society—all their greatest artists and scientists spend so much time sleeping that their rate of creation slows down significantly.)
In Peter David's Knight Life trilogy, Percival drinks from the Holy Grail while healthy, and is thus doomed to eternal life. When Arthur returns in the present day, he finds Percival drunk out of his mind in a New York City slum. He quickly whips him back into shape and made into one of his advisors. (And a good thing, too, because the Holy Grail is a big plot point for the last two books in the trilogy).
Coleridge's Ancient Mariner brings a curse upon his ship, and the crew spends weeks adrift at sea. Death appears to claim the crew, while the Mariner is claimed by a woman called the "Nightmare Life-In-Death." He lives through excruciating pain and horror, and considers death to be a relief. Although not explicitly immortal, he is seen as an old man who spends the rest of his life telling his cautionary tale to anyone who crosses his path.
Tom Holt's Flying Dutch. Most of the main characters are Vanderdecken and his functionally indestructible crew, forced to sea by the horrible stench that hangs around nearly all the time thanks to a dodgy elixir of life. One of them has adopted a hobby of regularly throwing himself off the top of the mast in the hope that this time it'll work. (All it usually results in is extra work for the ship's carpenter.) What really gets them isn't so much the immortality as it is the fact that they have to spend eternity with each other while stuck in the middle of nowhere with absolutely nothing to do for eighty three months out of every seven years.
Phil Plait, the Bad Astronomer, wrote a short-short story about the problem with living forever, titled "Deep". (The title is a reference to the cosmological concept of deep time.)
In Abraham Lincoln: Vampire Hunter, good-vampire Henry Sturges gives Abe a lecture on the disadvantages of being immortal, to explain why so many of his kind commit suicide in their third centuries. "Without death, life (becomes) meaningless. It is a story that can never be told. A song that can never be sung. For how would one finish it?"
Moral kind of broken at the end of the book, when Henry brings Lincoln back as a vampire after his assassination.
In Brent Weeks's The Night Angel Trilogy, the character of Durzo Blint is given immortality - with a catch. Durzo can still be "killed" in battle, but he always resurrects - with the twist that for every "death" he comes back from, one of his loved ones will die in his place. Over the course of seven centuries he turns from a Knight In Sour Armor into a bitter, sociopathic assassin.
This is discussed in Octavia Butler's Fledgling. In the book, vampires (or Ina) can prolong the life of humans they frequently bite (but they cannot turn them into vampires). Vampires themselves live long lives, but this is usually not a problem because they tend to keep to themselves. The humans often have problems, though, because they have to explain to their family why they look so young.
The short story "Divided by Infinity" applies this to the entire human race—it uses the many worlds hypothesis, then asserts that the human soul/consciousness/what-have-you is not destroyed when a given version of a person dies, but is instead transferred to whatever versions remain alive. Nobody ever dies, no matter how much they Wangst about it—they simply become less and less probable. (Not even suicide works, since there's always one version that chickens out.)
In German author Wolfgang Hohlbein's Die Prophezeiung (The Prophecy) an Egyptian pharao curses a traitor with immortality for killing him. We're not shown exactly what happened in the meantime, but 3300 years later all he wants is to finally die. As that would also mean death for all the Egyptian gods (being kept alive by the last person to believe in them), they are divided in those wanting to help him and those wanting to prevent this.
In the first century Satyricon by Petronius, Trimalchio, a secondary character, tells of visiting the Sibyl of Cumae:
For I myself saw the Cumaean Sibyl with my own eyes, hanging in a basket, and when the boys asked her, "Sibyl, what do you want?", she answered, "I want to die."
T.S. Eliot used the entire quote, in the original mix of Greek and Latin, as an epigraph to "The Waste Land".
One of the major plot arcs of the Deverry novels is the life of the wizard Nevyn, who in his youth swore an oath that he would not rest until he had set right the mistakes that had lead to the deaths of three of his friends, and found that the gods decided to make him keep that promise. As part of this, he frequently has to encounter the reincarnations of people he had met decades before, watch them die, and then meet them again decades later. The original three people his extended life centered around he runs into rather frequently over the course of his roughly 500 year lifespan.
One of the main characters of the Tide Lords tetralogy is an 8,000 year old immortal who is tired of eternity and wants to die.
Rock of Warrior Cats has been revealed to be immortal, but unable to interfere with the world.
Odysseus rejects immortality when Calypso offers it as an incentive to stay with her. (Somewhat ironically, in one of the non-Homeric continuations of his story, his wife Penelope and his sons Telemachus and Telegonos are all made immortal by Circe after Circe's son Telegonos accidentally kills his father).
Succession has the 'pink' faction in the Senate, who want to stop use of the symbiant and let folks die naturally. Several different factions and parties exist within this group with differing reasons for wanting the symbiont gone. Notably, some want this because they think it's better for society, as well as thinking that immortality is bad for individuals.
In the science fiction novella "Aqua Vitae," turning immortal is only the beginning of the protagonist's problems...mostly because if you don't have your life in order in the first place, expanding your lifespan indefinitely won't make things any better.
Discussed in Star Trek: The Buried Age. Data raises the issue with Ariel, a member of an immortal species. He points out that literature in many cultures explores the possibly unbearable tediousness of immortality. Ariel responds that to her people, life is too full of variety and opportunities to connect with others, and they have no issue with their non-aging status.
The title character of the Mediochre Q Seth Series has an insane Healing Factor which renders him trapped with a fifteen-year-old body indefinitely - he's about 400 at the time of the series. He doesn't like being immortal much, and he certainly doesn't like spending his immortality in mid-puberty. He's also possibly a little bit insane, and it's implied that this eccentricity is a coping mechanism. Worse, however, is his friend Melz, whose lesser Healing Factor rendered her trapped in a slow-aging body that's approaching 100 and - despite being blind, deaf and wheelchair-bound, isn't dead yet.
Firebird: The Katschei never seems to eat his magical fruit, or despoil his maidens (he more makes them dance and sing so he can prove he has power, not because he lusts for them), or even enjoy anything. It would appear that having his heart ripped out (selling his soul) removed his ability to enjoy anything other than power.
In L. Jagi Lamplighter's Prospero's Daughter trilogy, Prospero and his children seem to handle their immortality on the whole. They do, however, suffer anguish at the loss of beloved spouses and children, and the issue of maintaining identities has recently grown much more difficult. Refusing Paradise is a burden to Cornelius; he must return to blindness and hard work.
In Erin Morgenstern's The Night Circus, while the two men who start the competition appear to have warded off the boredom issue with such games, the discovery that they do not age (except the twins) has a heavy impact on those involved in the circus, albeit mostly concentrated on hiding that they do not age.
Explored in Elantris. Ten years prior to the start of the book the Elantrians changed from virtually immortal magic-wielding gods to withered people who couldn't die and couldn't heal from any wound. Any injury they took would remain with them forever, including the pain. The accumulated pain and hunger from mere existence eventually becomes too much for their sanity, eventually lapsing into a catatonic state.
The "devil" of The Damnation Game has a bad case of this, particularly lamenting that immortality has only made his terror of death worse.
Somewhither: The world of Cainem is inhabited by people who were never chased out of the Paradise, and thus never die. This is a very bad thing, since all earthly pleasures will eventually grow stale, while earthly sufferings never get easier. The only thing that slightly excites them is sadism and committing atrocities.
In Dora Wilk Series, vampire Eryk has almost been Driven to Suicide by the fact that after centuries of existence, his non-life has no purpose beyond drinking blood and hiding from the sun.
In Noob, Moulinof mentions that he knows how to make a potion that would make him immortal. He however refuses to drink it because he'd rather die than spend enternity in the same world as his Insufferable Genius former students.
In the Ray Bradbury short story "Homecoming", Timothy— the one young mortal in a family of immortal supernatural beings— is told this by one of his uncles:
"How much better things are for you. How rich. The world's dead for us. We've seen so much of it. Life's best to those who live the least of it. It's worth more per ounce, Timothy, remember that."
Warhammer 40,000 has the tagline of "In the grim darkness of the future, there's only war and the laughter of mad gods". One of the influences behind it, the Eternal Champion Cycle books, where a certained doomed soul is cursed to always be reborn to endless warfare and struggling against mad gods, only to be eventually be killed and reborn for more of the same. The one small mercy is the Eternal Champion doesn't remember their old lives, except for one incarnation who was punished for wiping out humanity.
In Timothy Findley's novel Literature/Pilgrim the eponymous character makes it clear how desperate he is to die when he is brought to the Burghozli Clinic in Zurich, Switzerland ca. 1912.
Celica in Akashic Records of Bastard Magic Instructor has this attitude. When her immortality was discovered, it caused most other people to fear her, with her lover at the time calling her a monster and running away. She did have some friends over the centuries, but they would inevitably die while she lived on. She tried killing herself multiple times, only to be prevented by a voice in her head that told her she had a mission to complete.