Keep in mind Vegeta primarily wanted immortality because it meant even if Freeza beat him within an inch of his life, he could crawl back and try again. With Freeza gone, he didn't have a want or need for it.
Fullmetal Alchemist makes this the prime motivation for Greed, Ling Yao, and May Chang. While Ling and May want to bring the secret of immortality back to Xing in order to obtain the position of Emperor/Empress for their respective clans, Greed is just, well greedy.
Smug Snake Kurt Godel of Mahou Sensei Negima! is this. When a child who recently became immortal is being discussed, everybody else focuses on the "will outlive loved ones, probably won't be able to have children, is quite possibly going to be in early puberty for eternity" aspects, but Kurt zeroes in on the "can become a king and rule forever with zero fear of assassination" aspect.
This is one of Orochimaru's biggest motivations. His obsession with immortality drove him to create jutsu that can raise the dead and another to possess the bodies of others. Strangely this is only a means to an end. His real ultimate goal is to learn every jutsu. No, seriously; he needs to be immortal because there are far too many to learn them all in a normal human lifespan.
Sasori, who turned himself into a human puppet in order to attain eternal life and an undecaying body as part of his philosophy that "true art" resists the passing of time (ironically he was also the first bad guy to be killed in the second season, a fact his partner, Deidara, who in turn lives by the completely opposite philosophy, lampshaded).
Kakuzu, a ninja in his early nineties who can extend his own lifetime indefinitely by stealing and replacing his own old, worn out organs with younger, healthier organs from people he defeats,
The goal of at least one character in each volume of Phoenix is this.
From Magician, This is pretty much what most of those chasing the immortal wizard Edermask in the hopes of getting him to divulge the secret to immortality are. Ironically Edermask's goal is to find out this secret too.
Desparaiah in Yes! Pretty Cure 5 is one of these. She succeeds, and explains that she did so because she didn't want to grow old, but she still feels unhappy and filled with Despair, being her namesake.
In the Fairy Tail movie The Priestess of the Phoenix, this is the goal of the villains through attempting to capture a phoenix. One of them, Dyst, has a fairly silly or tragic (depending on your point of view) reason for wanting immortality. As a child, his pet weasel died and he tried to bring him back. Upon finding out that there are no spells to bring back the dead, Dyst became terrified of dying and vowed that he would find a way to live forever. His quest eventually turned him into a vile, twisted individual.
Hob Gadling of The Sandman. Overall, he's glad he's lived so long, and he sees that the world is getting better.
An obsession with becoming immortal was what drove DCU Mad Scientist Professor Ivo's early schemes. Then he got what he wanted, unfortunately.
Batman foe Ra's Al-Ghul is either this or Heir Club for Men. Sometimes both. At his worst, he combined the two to try to claim a fresh young body for himself. He's staved off death for centuries via the Lazarus Pits, but his ultimate goal was to find a way to cheat death permanently.
Bella in Luminosity wants to live forever, and meets a vampire. Interest ensues. Ultimately, she'd like this for everyone.
Tom from The Fountain is a doctor searching for immortality.
Star Trek: Generations: Soran would have effectively achieved this if he has re-entered the Nexus. In a way, he did anyway. Since the Nexus exists outside of time, he's still there (and always will be) even though he was also killed outside of it.
Lord Voldemort from Harry Potter, due to a pathological fear of death. He's also adamant that only he should be allowed it, at one point stating "Only I can live forever" right before he kills Snape. In this series, you have to kill people to be immortal. This allows you to split your soul and hide parts of it, guaranteeing that you remain earthbound if you were to die. This ultimately resulted in a particularely karmic fate for him. Due to the fact that his soul was still split up when he did in fact die, he wasn't able to pass on to the afterlife, nor return to the world of the living. His life-long flight from death resulted in him being trapped in an empty limbo for the rest of eternity.
Only some methods of immortality require murder. The first book talks of Real Life legendary alchemist Nichoas Flamel, and concerns Voldemort's efforts to steal the Philosopher's Stone that allows making an elixir for immortality (though probably only Agelessness). The stone is destroyed at the end to prevent Voldemort getting his hands on it, with Nicolas and his wife Perenelle accepting death after living happily for centuries.
In Larry Niven's story "Cautionary Tales," a human looking for a way to live forever goes to the center of the galaxy and runs into an alien looking for the same thing. Tales of living forever are in all cultures, but only humans have "cautionary tales." The alien has been looking for far longer than the human...
Also the Undying Prince, though they use very different methods- Tarrant is an Emotion Eater, while the Prince practices Grand Theft Me though he keeps his original body in a vegitative state in a tank- he needs it as an anchor even if he's not using it anymore.
In Journey to the West, Sun Wukong responds to almost every piece of advice from his mentor with (paraphrased) "Yes, but will it make me live forever?"
Robert Silverberg's The Book of Skulls. All four of the protagonists are looking for eternal life. Which ones are the villains and which the heroes for doing so becomes increasingly less clear-cut as the novel progresses.
Bella Swan from Twilight is called this by some readers of the series.
In Michael Moorcock's Dancers at the End of Time saga the humanity has reached immortality with little cost (at least immediately apparent), and consists of a few hundred near-omnipotent individuals who mainly seek to have a good time, having given up the old morals and social standards as useless, since nothing they do can actually harm anybody else in any significant sense.
This is the goal of the Howard Foundation in Robert A. Heinlein's "Future History" timeline. Founded by a rich man who found himself dying of old age in his forties, it embarked on a program of human eugenics (before genes were even understood) by the very simple method of paying people with long-lived ancestors to marry and have children. Hundreds of years later, this eminently practical program produces humans with more than double the typical lifespan. Forced to flee Earth on an experimental spaceship due to public jealousy, the Howards return decades later to discover that, in their absence, humans have invented treatments that can prolong life enormously. Multiply this by the Howards' inbred longevity and you have a recipe for near-immortality. The longest-lived human, Lazarus Long, is nearly 2,500 years old by Time Enough for Love and shows no signs of stopping.
In The Secret History, Julian argues that this is what every human secretly wants, and it becomes a recurring theme throughout the book. Too bad Bunny ends up murdered, Charles tries to kill Henry, Henry kills himself, and Francis attempts suicide.
Discworld has Alberto Malich, a wizard who endeavored to become immortal. This is especially troublesome in this setting, since all mortals have a predetermined, finite amount of time to live and someone not dying while they are supposed to can tear all of reality in two. It did work out for Alberto after a fashion, when he became Death's personal servant, now known simply as Albert.
Darth Bane attempts to gain immortality by continuously transferring his soul from body to body as they became old and frail.
Subverted with Etheldredda, as she takes the immature potion of immortality that makes her only a Substantial Ghost and gets eventually destroyed by Marcia Overstrand in the end of Physik.
Doubly Subverted with Marcellus Pye, as he first makes a potion without a critical component that gives only Age Without Youth. Septimus finally makes the complete potion and passes it over to the ailing Marcellus.
The Alex Benedict novel Polaris has the scientist Dunnager, who has made it his life's work to find a way to halt the ageing process.
Examined in James H. Schmitz's The Demon Breed when Ticos Cay explains why he does indeed want to live forever. (Or to start off with, at least 1000 years).
Marcus of Babylon 5, when hearing the phrase "Who wants to live forever" when about to undertake something dangerous, responds "I do, as a matter of fact." Ironically, he pulls a Heroic Sacrifice.
Doctor Who: the Master and Borusa, and a few one-off villains like Lady Cassandra and Professor Lazarus. It doesn't go that well for any of them, since "everything has its time" is a recurring theme (especially in the revival). The Master gets away with Joker Immunity. Incidentally, the novels give one of the Master's pseudonyms as 'Koschei' - as in Koschei the Deathless listed above.
In TorchwoodAngelo is one, mainly because he wants to spend the rest of his life with Jack.
Richard Alpert on LOST makes a deal with Jacob to live forever because he's terrified of dying and going to Hell because he accidentally murdered a man. Eventually, he comes to regret his choice.
House of Anubis has the Victor and his society, as well as Rufus Zeno. Victor's father had been after it as well, which meant Victor's basic goal was to finish his father's dream. Other members of the society had different goals as well; While most were selfish, it is known that Jason was dying of a degenerative illness, and would have died young unless he obtained the immortality. It is unknown what Rufus's larger goal was, but he was certainly more ruthless about it, willing to kill people if it got him what he wanted.
Of course, at least in third edition one wonders why they bothered - there are several other kinds of immortality easily reachable (in game terms) that don't involve leaving you as a rotting corpse.
Unfortunately, even with magic, being a lich is only a half-measure of immortality. DnD liches have to contend with their sanity and intelligence decaying away over the one thousand years that basic lichdom lasts. After that they are nothing more then a floating skull, or Demilich. Demiliches are beings of absolutely incredible power, having lived long enough to learn every secret of magic that ever existed, but are also without exception batshit insane and utterly consumed by mindless insanity and loathing for everything. As far as immortality is concerned, Demiliches are virtually impossible to either destroy or permanantly kill. But, ya know, the whole 'mindless insane bodiless skull forever' might be a bit of a downside.
Depending on which sourcebook you're reading at the moment, anyway. There are very nearly as many variations on lichdom and the lichification (is that a word?) process as there are books in the D&D line of tabletop RPGs. One variation, for example, requires zero babies, but instead a ritual involving the heart of a sentient humanoid that must be performed every 100 years. There is no obvious rule why this would mean a human heart instead of an orc or troll, or a convicted murderer. The same sourcebook says that demiliches are so decayed because they spend all their time traipsing through other planes of existence via astral projection or some such thing (whether they're likely to be insane after who knows how many millennia of existence lies in the eye of the beholder).
There is a less evil and more described way to become a Lich described in the Power Class: Alchemist mini-supplement.
There are a breed of Liches known as Baelnorn Liches who are exclusively elven, and limited only to non-evil alignments. Unlike most Liches, they do not do what they do in order to live forever to gain power. Instead, through a divine ritual, their immortality is gained by swearing to become an eternal protector of elves and their lands.
In OA7 Test Of The Samurai, the evil Za-Jikku tries to become immortal by changing the world's atmosphere to a substance that will let him live forever. Unfortunately, breathing it will kill all other creatures who haven't prepared as he has.
So he's basically trying to turn the air into shinsoo then?
On top of any drawbacks whichever method the above chose, there's also the problem that the inevitability of death is supposed to be an universal law, and those laws have enforcers. In this case, huge, armored humanoid robots that'll get briefed as to how to undo this immortality, find the offender, chase after him relentlessly and beat him to death until they're sure he's not coming back.
In GURPS, there are some spells that can "steal youth," take months off your life, or halt aging. They are generally so expensive and limited as to not be worth it (the potion version of the Youth spell takes almost a year to make, any failure in making it causes the user to age faster, and it only takes one year off your life.) If permanently enchanted on a wearable item, the Halt Aging spell has such a ridiculous energy cost that even a Great Wish won't be enough to make one. (The book notes: "Kingdoms have been toppled for possession of such things...")
In the powered by GURPS Scenario Transhuman Space you can become immortal by uploading your mind into a computer (if you have the money), though your biological body will die in the process, or before.
Zouken from Fate/stay night, who originally prolonged his life to reach a goal but later went insane and forgot why he did it in the first place. Failed to become a vampire and thus suffered the "immortality without youth" drawback of Who Wants to Live Forever?.
In Prince of Persia: The Sands of Time, the Big Bad suffers from impending death from "consumption" (tuberculosis), and seeks eternal life— at the expense of the world. In the game's end, the Big Bad offers to share eternal life with the hero— who of course refuses it, in order to turn back time and bring everyone back to life.
In Tales of Symphonia it is revealed that Mithos Yggdrasill and his companions found a way to stop individuals' biological clocks, essentially locking their bodies in the age they please. At the end of the game, it is revealed that he intended to convert all living beings in his universe to this state in order to reduce discrimination brought about by humans against half-elves for their human-like appearance and their naturally long life cycles.
Kaguya Houraisan ordered her vassal to concoct an elixir of immortality on a whim and drank it on a whim, leading to her exile from the moon. As a Lunarian, Kaguya was already ageless, but now she's immune to all other forms of death as well. Said vassal, Eirin Yagokoro, might have also drank the elixir, but that's less clear.
Fujiwara no Mokou (who was previously an ordinary human) also drank the same elixir of immortality, some of which Kaguya had left behind as a gift to the Emperor of Japan. Mokou stole it from the soldiers tasked with throwing it into a volcano and drank it in a moment of weakness, something she's regretted ever since. Well, moreso the murder of the soldiers than the immortality.
Protagonist Marisa Kirisame has been known to perk her ears up at mention of easy routes to immortality but so far hasn't had the patience or the nerve to follow through (one of her options would have been to eat Mokounote Or Kaguya. But Marisa didn't know this yet when she fought Kaguya.). She's also rejected several forms of immortality: she's specifically trying to keep her humanity while also being immortal, so any method that turns her into a youkai is out.
Tenshi Hinanawi achieves de facto immortality by beating the crap out of each shinigami that's sent to claim her.
The Buddhist priest Byakuren Hijiri abandoned her teachings after her brother died and instead desperately pursued immortality and eternal youth, which she managed to achieve through black magic and becoming a youkai. She later rediscovered Buddhism though and therefore presumably doesn't feel the pressing need to preserve the immortality she already achieved.
As well as a number of other characters who actively chose to become immortal during their lifetimes, although less emphasis is put on that as a motivation (Alice, Kasen, Seiga, etc).
In Marathon, this is Durandal's motivation. Yes, he's an AI and therefore already technically immortal so long as he keeps his hardware maintained, but he plans to outlive the universe and transcend reality, becoming truly immortal in every sense of the word. He fails... though he does live to see the universe's natural end.
Xanatos of Gargoyles fame. What's all the money and power in the world (of which he has quite a bit) if he can't enjoy them forever? He's tried a number of things but never desperate enough to use them without testing on someone else first.
He eases up on this after Hudson reminds him of Demona and Macbeth, who actually are immortal and fairly miserable, and follows this up by asking him an Armor-Piercing Question concerning the legacy Xanatos will leave behind. Xanatos eventually decides that his family is more important.
The pursuit of immortality is a perennial pursuit in mysticism. Western alchemists spent their lives seeking an immortality potion that was variously called aqua vitae, panacea, elixir, the philosopher's stone, or literally hundreds of other names.
As did Eastern alchemists, including Chinese Taoists. Their elixirs tended to be based on gold, mercury and other heavy metals, so the effect might have been more pickling than life-extending.
There were also some scriptures that recommended a kind of sexual vampirism to keep practitioners young. The general idea was to choose very youthful partners and sometimes to switch in the middle ...
Their is some evidence to suggest that GUN POWDER actually was one such elixir, ironically.
One of the primary goals of transhumanists today, through advances in science.
Note that not all transhumanists desire immortality, though most do. Also they discuss the ramifications of an unlimited lifespan regularly and the general consensus is that the only cost would be that you'd effectively cease to be "human" (a fair price many think).
The real life Ponce de Leon averts this trope. He went to Florida in search of gold and to expand the Spanish empire. Only after his death did wild stories of his search for the Fountain of Youth begin to appear.
Arguably, this is the goal of most mainstream religions. "Everlasting life" is usually somewhere among the reasons why people choose to follow them.
Christianity teaches that immortality was the starting condition, it was forfeited by Adam and Eve.