Saint Seiya has the heroes fight Hades, god of the underworld... and his minion gods Thanatos and Hypnos, gods of death and sleep, respectively. Just how powerful they are or how wisely they use their abilities is debatable, as they both physically fight the saints (and lose) rather than kill them by natural causes or make them comatose, respectively.
In Hypnos' defense, his special move does make the target go into an endless sleep, with the caveat that it has No Ontological Inertia once he is beaten. Thanatos, on the other hand, may just be an arrogant Jerkass who prefers toying with his opponents rather than plain killing them.
Thanatos tries to use the insta-death on a non-fighter (Seiya's long lost sister) but the lowest ranked saints manage to block it. It's implied that the nearly godly main characters would be immune.
After The Mighty Thor severely ticked off Hela (Norse goddess of the dead) one time, she cursed Thor to be undying, but to have extremely brittle bones, leaving him in constant pain and held together by an improvised suit of armor and splints. (Obviously, he got better.)
When Deadpool and Death become infatuated with one another after Deadpool has a number of near-death experiences, a jealous Thanos prevents Deadpool from dying and joining the entity.
Inverted in Universe X. Death is under siege by Captain Mar-Vell's army, and fights back, but it seems she'd be perfectly willing to leave him alone if he'd return the favor. And then it turns out that the world needs a Death.
The 2009 Marvel MAX destroyer series by Robert Kirkman and Cory Walker featured the elderly hero refusing to go with the Grim Reaper squad into the afterlife. One of them makes the mistake of threatening to take his family if he doesn't come willingly.
Shuma Gorath and the other Many-Angled Ones alter reality in the dimensions they take root in to remove Death so that life will grow out of control and become cancerous, spawning more Eldritch Abominations like themselves. And even Death itself (as in the cosmic anthropomorphic personification of death) can't permanently kill them.
Marvel's Elders of the Universe were essentially immortal already, but after they offended Death, she refused to accept them, forcing them to live on, no matter how horribly maimed.
The Final Destination films have the protagonists fight Death in the form of a faceless Butterfly of Doom that's out to get them for not dying when they should have. Except it is revealed (or retconned) in the fourth movie that the characters DID die when and how they were supposed to- the visions that saved them from dying were sent by Death itself! Suck on that, causality!
Harry Potter and the Deathly Hallows has a fable that takes this form: three brothers use magic to cheat death, and Death himself shows up to "congratulate" them with three gifts. Two of the brothers die soon after because they chose or used their gifts poorly, just as Death hoped, but the third uses his gift to hide from Death, and decades later, as an old man, he decides he's lived long enough and passes the gift down to his son. It's suggested that this gift is Harry's invisibility cloak and that Death has poorer eyesight than a cat.
The Auditors of Reality, on the other hand, who are roughly personifications of the laws of the universe, are antagonistic towards all life, especially intelligent beings. They have it in for Death, too, particularly as he takes the side of life against them; this results, especially in Reaper Man where he's a protagonist, in the odd situation that Death is "Enemies with Death".
In Hogfather, Mr. Teatime devises plans for killing Death, as well as seemingly every other anthropomorphic personification on Discworld. In his spare time. And nearly pulls off the one against the Hogfather (a Santa Claus expy) until Death and his daughter Susan get involved.
In the Fafhrd and the Gray Mouser stories the Death of their world is something like a cosmic Bureaucrat that answers to some vague Pan-dimensional forces. He has to meet particular quotas of who dies within allocated time periods, measured in mortal heartbeats, and is only really allowed to do so by manipulating natural events. He can't snap his fingers and kill somebody unless they have a severe medical condition but he can make sure that they might have a nasty incident with some badly maintened masonry. The exceptions to this rule are the protagonists who once attempted to steal his mask, since then he's had it in for them and has resorted to toying with reality by teleporting crazed berserkers into thier living quarters.
In the Malazan Book of the Fallen a legendary Malazan general started out as a disciple of Hood, the God of Death but felt betrayed by his god when his beloved daughter died. He declared himself an enemy of the god and spent decades pursuing a way to kill Hood. Hood himself really does not have any strong feeling on the matter.
In the Iron Druid Chronicles the protagonist has the inverse problem. A female goddess of Death of his religion is rather fond of him and likes to use him for sex. This has given him a form of immortality but has also made him a mortal enemy of one of the gods of Love. Since in that pantheon all the gods are very nasty, vengeful and quite barbaric, the difference is mostly academic. The god has tried to make the protagonist's life miserable for two millenia and is always looking for a way to kill the protagonist.
In The Tiger's Wife, by Téa Obreht, the narrator's grandfather tells of his encounters with the "deathless man," who angered Death — his uncle — by saving a woman who was meant to die, and was punished with immortality.
Live Action TV
The finale of Red Dwarf has Rimmer escape from Death by hitting him with a Groin Attack. Made even funnier if you flash back to Series I, when Lister picked up a length of pipe to ward off his inevitable demise:
Rimmer: You can't just whack Death on the head! Lister: If he comes near me, I'm gonna rip his nipples off.
The demon Alastair uses Death's scythe to kill two Reapers, part of a ritual to break one of the 66 seals on Lucifer's cage. He's stopped by the Winchesters from killing Tessa and completing the ritual though.
In the later part of season 5, Dean hunts down Death so he can kill him and take his ring. When he finally meets the Grim Reaper, it turns out that things don't work that way in the Supernatural universe.
In the first episode of season 7, Castiel and Death get into a very heated argument, in which Castiel outright threatens to kill Death. Death isn't impressed by the "mutated angel".
The Outer Limits (1995): In the episode "White Light Fever", Harlan Hawkes is a billionaire centenarian with an abject fear of death due to his extremely traumatic childhood experiences, and uses his wealth to reserve revolutionary medical treatments for himself. The Grim Reaper concludes that he's outstayed his welcome, and starts to hunt Hawkes Final Destination-style.
Pathologic has an odd version of this. One of the main characters is a Bachelor of Medicine who honestly believes that the very concept of "death" is not a natural part of our existence, but rather another disease we have simply yet to cure. Some of the NPC's even refer to death as his natural enemy, which is true on more than one level given his profession.
A significant theme in Persona 3. And ultimately, all you can do is hold Death off for a time - the battle is not truly winnable. In a less existential sense, The Grim Reaper will hunt you down and (unless your levels are significantly stacked) wipe the floor with you if you dawdle too long in Tartarus.
In the first Shadow Hearts game, the God of Death embodied as Fox Face doggedly hunts Yuri through half the game, apparently having a major grudge against him that is never spelled out, beyond wanting to help the monsters killed by Yuri take their revenge. Implied to be one of the innumerable lesser death gods as common in Japanese mythology, as opposed to the ultimate personification of Death as in Western mythology.
Twisted Metal 2's Mr. Grimm is in fact the Grimm Reaper. He's pretty weak but has the single most powerful projectile in the game. His reason for entering the tournament is to make people die faster to feed his hunger for souls. He gets his wish.For a while. When global war breaks out, its only so long until everyones dead and there's no more deaths, ever again.
In Dantes Inferno, the protagonist Dante is assassinated. When Death appears before him and tells him that he will be damned for his sins, Dante gets up, takes the knife from his back, and fights Death. Dante won. But not before stealing Death's scythe and killing him with it.
And then it turns out Death never came for him in the first place as a tangible being. He just keeled over on the spot.
Happens in Touhou 7: Perfect Cherry Blossom when you fight Yuyuko, a super-powered ghost whose main power is inflicting death upon others. Said battle happen in the local equivalent of afterlife. A more literal use of this trope is during Touhou 9: Phantasmagoria of Flower View when you must fight Komachi to get her back to work, so it's your character who deliberately make Death your enemy.
What is important to remember is Komachi is just a ferryman shinigami who takes the spirits over to meet the Yama, not a reaper shinigami who comes to collect the person when their time is up. However, Tenshi the Celestial is immortal through the fact she has defeated every reaper shinigami who has come for her, which makes the fight against her in Komachi's storyline a Crowning Moment of Awesome.
Hermits are also on probation from Hell for their immortality, and have to perform good deeds or get taken by a shinigami. Hermits who don't, like Seiga, have to fight off or escape the servants of Hell to survive.
Fallout Tactics has this as a potential character trait for Ghouls. They gain access to perks faster than regular ghouls, but they have a chance of taking damage for no reason other than DEATH!!!
Chakan is based on this. A Swordsman brags so much about his skills that he declares not even Death could better him. Death accepted the challenge with a condition: If Chakan could defeat him, he'd be granted eternal life. If Chakan was defeated, he'd become Death's eternal servant. Chakan wins, but turns out he was Blessed with Suck since he cannot rest until all Evil is eliminated.
In Odin Sphere, several of the characters have to fight Odette, the Lady of Death, for various reasons. Gwendolyn succeeds in killing her.This wound up being something of a mistake, since it winds up triggering the apocalypse, killing all but four people on the planet.
Although the Queen of Maggots in The Cat Lady is, by her own admission, not truly "death", she may as well be given her ability to resurrect the dead and prevent souls from passing on into the afterlife. The game provides an excellent example of what happens when you annoy Death: she picks a recently-deceased soul, resurrects them, and makes them functionally immortal while telling them that the only way they get out of their crapsack life is by murdering the hell out of you.
The Fire God Agni from Kubera fights the god of death just to save Brillith.
Skips in Regular Show. Also arguably the whole group as of "Skips Strikes," since Rigby won a bet against Death and defeated Death's bowling team.
Another borderline example is in "It's Time" where Father Time is shown to be frustrated with Mordecai for microwaving his clocks and decides to free him from eternal imprisonment only because he accidentally trashed Father Time's living room.