Many cultures have the idea of an eternally recurring hero. Not a King in the Mountain, just a hero who keeps coming back for more. Maybe it's one hero with multiple identities. Either they're immortal, or there's an element of returning from being apparently dead or seeming way too old to fight. The reason both are included is that this trope is as old as mythology, and in its early era the concepts of immortality, resurrection, and absurd longevity were fairly interchangable.
A character could become an Eternal Hero for many reasons. Maybe, like the Irish legend Osinn in Tír Na nÓg, the hero ends up in a time loop or Neverneverland that allows him to return to Earth centuries later without having aged. Maybe he's just unkillable. Sometimes, a deity or other force of nature embodies itself as this character whenever the world needs it. For the Norse, Tolkien argues that it's Sigurd/Siegfried. For the Celts, it's the many permutations of Fionn and the Fianna and the Red Branch Knights.
Supertrope for The Hero with a Thousand Faces, the King in the Mountain and the Legacy Character as well as many heroic archetypes. Sister Trope to Immortality.
Sandman also provides an extreme example, where the death of one of the Endless automatically means the nearest suitable human turns into them. Not to mention the immortal Hob Gadling, or Death's assertion in Dream Country that gods and mythologies "live on in a kind of dream country" long after humans have stopped believing in them
As of Whatever Happened to the Caped Crusader?, Batman definitely counts as a multiversal version. The idea, manifestation, and embodiment of Batman is inevitable in any timeline. It's implied that all the Batmen in different universes have or will reincarnate into each other.
In fact, that entire comic is pretty much one big depiction and analysis of this proposed trope, as a side affect of trying to be the end-all be-all summation of Batman in all his forms. (It was written for when Batman died in canon, so it's a eulogy of sorts)
Mouse Guard: Fall 1152 references this at the beginning of Chapter 4 with a quote attributed to the Black Axe:
"Death is as powerful a weapon as it is an easy escape. Heroes can pass in to legend, Legends into myths, Myths fuel new heroes."
About half of the huge cast are Detian, which means that they've taken the Omega-2 retrovirus which gave them The Ageless with a moderate Healing Factor. Began as heroes whose deeds included saving the Autobots from destruction and leading a revolution on the planet Zardon. They then were scattered by the Big Bad's Batman Gambit, and went into exile for 100 years. They came back and reconstituted the Wedge Defense Force in the 2380s, just in time to save the whole of the Perseus Arm of the galaxy from coming under the rule of the GENOM corporation.
Utena Tenjou is the Rose Prince of Cephiro, a recurring office given to winners of the rose duel tournament. While the Pillar of Cephiro holds the world together and keeps it peaceful by will and prayer alone, the Rose Prince is a roving correction mechanism whose fate is to always be a champion for people in need. She plays a large role in the defeat of the Earth Alliance and Psi Corps. I don't think she's actually immortal yet, but given her close relationship with the Aesir, it's bound to happen eventually.
Michael Moorcock's The Eternal Champion series (The Elric Saga, The Books of Corum, Von Bek, Count Brass, et al) is one of the main modern literary examples. It's an epic series covering over 40 books and almost as many individual incarnations of the titular champion and moves between straight High Fantasy and Science Fiction.
At the end of Greg Bear's City at the End of Time, it's revealed that Daniel is actually Sangmer, the legendary missing-presumed-dead hero that the characters in the scenes set several trillion years in the future read stories about. When the entire multiverse started to unravel during his lifetime, his demiurge Eternal Love Mnemnosyne regressed him to childhood and sent him to be a King in the Mountain in a Pocket Universe. Eventually he is released, with his memories of being Sangmer suppressed, as a human fate-shifter (someone who can jump between parallel universe versions of themselves to avoid bad luck). He then starts from the beginning of human history, journeying to the recent past, where he has shifted into the identity of Daniel, who in turn shifts between multiple Daniels until he ends up in a universe where he is a beggar called Charles Granger. It turns out that this is because Daniel died as a teenager in this universe, so he ended up in the nearest equivalant. He then transfers his consciousness into the body of theoretical physicist Fred, his best friend in his home universe and married to the late Daniel's sister in Granger's world. He does all this, as well as being a Manipulative Bastard and The Sociopath, because his suppressed memories are driving him forward to the point where he can stop the multiverse's destruction and reunite with his love. Because she's a demiurge and he's a far-future descendant of humanity, they don't really care how many human identities he sacrifices to succeed.
Cohen and his henchmen do fit the trope. At the end of The Last Hero, they suffer a huge explosion that should've killed them. But Death doesn't come for them. Why? Because of this trope.
All of the protagonists of Kim Stanley Robinson's The Years of Rice and Salt go through serial reincarnation down through the history of an Alternative Universe Earth from the moment it branches off from real history to the AU's 'present'.
In Guy Gavriel Kay's The Fionavar Tapestry, Lancelot and Arthur are set up as eternal heroic archetypes who appear in all the worlds of the Weaver's Tapestry, playing out the same roles of the Noble King and the Knight Who Betrays Him in as many guises and names as there are worlds.
Parodied in Craig Shaw Gardner's The Wanderings of Wuntvor series- Wuntvor, former apprentice to the great wizard Ebenezum, evades the Anthropomorphic Personification of Death so many times, Death begins to think he is "The Eternal Apprentice", and gets really annoyed about it, the more frequently it happens.
The Dragon from the Wheel of Time is example. He's reincarnated once an Age to do battle with the Dark One. Also in the Wheel of Time are the Heros of the Horn, who reincarnate more frequently and form much of the mythology and legends.
Perry Rhodan: an interstellar hero who became immortal during his adventures.
The Doctor in Doctor Who is an archetypal example of this trope. It helps that he has a time machine that can go anywhere in time and space, so he literally can reappear at any time. His regeneration may make him a slightly different person each time, but he's always a hero.
The song "Outsider Intro" (quoted at the top of the page) by DJ Shadow features a sample of a storyteller describing a mythic figured known as "The Outsider"
But in the darkest hour
Whispers begin to tell of a figure emerging from the darkness
A being without a name, faceless and obscure
Part presence, part idea they say
As if the very force they describe has existed for eons
A dormant seed awaiting nourishment
Most ancient mythologies have an Eternal Hero. In Ancient Grome, it's those heroes who are semi-divine, like Heracles. For the Norse, Germans and Anglo-Saxons, it's Siegfreid and his Expy son/alter-ego Sigurd. The Irish have Oisinn, Fionn and Cuchullain. In other words, this trope is very much Older Than Feudalism.
In The Hero with a Thousand Faces, Joseph Cambell discussed the use of the archetype of the eternal hero in different ancient mythologies, positing that they might all be facets of a single ur-hero and ur-myth.
Latter-Day Saints believe that the Archangel Michael, who defeated Satan during the War in Heaven, is the same being as Adam, the first man on Earth after its creation. Adam being his mortal name and Michael being his heavenly name.
In the Legend of Zelda games, various boys called Link across three alternate versions of the same universe take up the mantle of the hero Link to save the world. However, they're not Legacy Characters, they're different incarnations of the same eternal hero.
The world of Avatar is based around four elements: water, earth, fire, and air. The world is roughly divided into four countries, each of which corresponds with a particular element. Certain people from each country can manipulate (or "bend") the element associated with their culture. These people are known as benders; bending gives them abilities significantly above and beyond those of normal people, but nobody can ever bend more than one element. The exception is the Avatar. Every generation, an Avatar is born to one of the nations; this person can learn to bend all four elements. They can also access the Avatar State, which allows them to channel the souls of all the previous Avatars to gain incredible power. Once their training is complete, they typically become an intermediary between the four nations, maintaining balance and ensuring that the world does not come to any great harm. Once the Avatar dies, a new Avatar is born to the next nation in the cycle. Avatars can talk to and get advice from previous avatars. Every Avatar has a different personality, but they all share a strong goal of protecting the world and the people around them.
Captain Planet, as the anthropomorphic personification of Gaia, probably counts.
Five hats means that five tropers think it is ready to publish.
You are saying that you think this draft is ready to be published. That means the description is not ambiguous,
it doesn't duplicate an existing trope, there are at least three examples, and the title makes sense.
Is that what you meant to do?
You are saying this draft has a ready-to-publish hat it does not deserve and you are taking it back.