"We conceive of immortality as having a beginning, but no end; but we conceive of eternity as having neither beginning nor end. Hence it is proper to speak of eternity as the attribute of God, but of immortality as the attribute of man."
― HORACE MANN, Thoughts
"Time... Time flows by us like water. The ages pass until we are weary, so weary of life. Everything we create is destroyed, the lands we love are upturned. Mortal man may pass beyond this world, even if they do not know what lies there, but the elves... the elves must stay. You are about to face something, Ancanar, that may require stronger faith then you now have. Death... death is a gift."
The user is absolutely immortal, and can not to die, age, get sick, be permanently wounded, is absolutely self-sustained and, as their mind and soul are as immortal as their biological bodies, they are immune to mental/spiritual damage. Any injuries the user suffers immediately heal, even if the user's body is disintegrated or blown up or completely erased and even if the user is completely destroyed to the sub-atomic level, they will still return to life. Alternately, some users are absolutely immune to all harm, and can never die from any natural causes.
The user's existence and soul become completely free and independent of even the concept of reality, making them unbound to the subjects of life, death, and manipulations. Also, the power cannot be removed, rendered truly perfect and absolute.
Ladylee Tangleroad (A Certain Magical Index) ate Ambrosia, which twisted her cycle of life force, granting her complete immortality that she could not get rid of.
As the Adam of Darkness, Dark Schneider (Bastard) can no longer die or be killed by anyone/anything except the Adam of Light, and not even by God can effect him anymore.
Sōsuke Aizen (Bleach) attained immortality and indestructibility after fusing with the Hōgyoku, which kept regenerating him no matter the damage, resulting in the necessity to seal him away deep in the underground prison.
Lobo (DC Comics) has caused too much damage in both heaven and hell that he has been permanently banned from the afterlife, rendering him completely unable to die, even when reduced to a puddle of blood that he regenerated out of.
Garlic Jr. (Dragon Ball Z) made a wish for immortality with the Dragon Balls, allowing him to regenerate from all bodily damage and never age nor die.
Future Zamasu (Dragon Ball Super) made a wish for immortality with the Super Dragon Balls, allowing him to regenerate from all bodily damage and never age or die...
...to the extent of instantly regenerating his head after it was blown off.
Porky Minch (EarthBound) abused time travel so much that he trapped himself in the current timeline, resulting his inability to age further nor die in anyway. He lived to the end of the world, and is still alive.
Bolt Crank (Eat-Man) is so immortal that not even Death could affect him in any way, despite Death physically manifested in an attempt to end his unnatural agelessness.
Kagerō (Flame of Recca) used the forbidden time-portal spell Jikuryūri (Time-Space Separation), which traps her outside of space-time, cursing her to become immortal.
Sun Wukong (Journey to the West) become unable to die or be harmed in any way after eating both the food of the heavens and erasing his name off death's register.
Raziel (Legacy of Kain) is the agent of the Elder God, and is beyond death; if he takes too much damage in the Material Realm, he will simply be forced back into the Spectral Realm, and if defeated in the Spectral Realm, he will eventually reform.
SCP-682 - Hard-to-Destroy Reptile (SCP Foundation) possesses regenerative properties so advanced that every attempt the SCP Foundation has made at ending his life has failed miserably.
Because Timmy Taco wished that he and Sally Sandwich (Sweet Story) would be "married forever", it made it so that neither could ever die - including immunity to the effects of starvation or thirst - the wish making sure that their marriage does not end through death due [they] part.
Ouroboros (Valkyrie Crusade) is the symbol/embodiment of eternal return, granting her true immortality.
Lich (Valkyrie Crusade) is absolutely immortal, as she is beyond life and death.
Chione (Valkyrie Crusade) is so obsessed with her own beauty, that she used magic to gain eternal immortality in order to stay beautiful forever.
Asphodelus (Valkyrie Crusade) is completely immortal as long as exists souls in the land of the dead for her to feed of.
I addition of being an ageless fable, Prince Brandish (Fables) performed a magical procedure that removed his heart, making him unkillable as long at the heart is kept untouched.
Death (Adventure Time)
Hunson Abadeer (Adventure Time) is a "deathless" demon, meaning he cannot be killed.
Due to being the gatekeeper of the Land of Shadows when it was sundered from the world while still a human, Scathath (TYPE-MOON) became unable to die by any means, forcing her to remain alive until both the World and Place Outside The World/The Root completely vanish from existence.
Morgan le Fay (Marvel) has been 'killed' more times and in more varied ways than any other villain in the Marvel Universe. Despite this, she has survived each and every single time, most often appearing seconds later after her 'death' seemingly unharmed.
Fujiwara no Mokou (Touhou Project) Self-Abandoning and Immortal Human.