Born-Again Immortality

In this type of Immortality, a character is not particularly invincible or unaging. They can be hurt and killed, and they usually age normally. However, if they should die, they will immediately be reborn as a child.

This child may or may not be identical to the original body, but will usually have all the memories of their past life or lives. Note that this is not always Reincarnation - the child may not literally be born, but instead appear or grow on its own.

If they "respawn" as an adult (or at the same age they died, if a child), then it's Resurrective Immortality, instead.

This type of immortality rarely overlaps with the other Immortality tropes. See also The Phoenix, Reincarnation.

Examples of Born-Again Immortality include:

Anime and Manga

All Digimon in most seasons have this, though Digimon Tamers averted it. When they're killed, they merely turn into a Digi-egg and are reborn, though it varies if they remember their previous life or not. A plot point in Digimon Savers is the Big Bad, Kurata found a way to prevent them from coming back to life after death.
Asakura Hao in Shaman King, who reincarnates himself every 500 years (having done this twice at the time of the series) to compete in the Shaman Fight. Could possibly reincarnate more frequently, just chooses not to.
Seems to be implied of all gods and goddesses in Ah! My Goddess. It's the only explanation that makes sense for how Belldandy, for instance, could be known to and worshiped by the Vikings, and yet be a child at the same time Keiichi was one.

Fan Works

According to supplemental materials for the fanfic cycle Drunkard's Walk, this is part of the powers of the Warrior called "Skitz", except all the individual incarnations' personalities remain separate from the current one, and they act as a Split Personality, trading off on controlling the body.

Film

A characteristic of Elora in Willow, necessitating a more complicated process of disposal.

Live-Action TV

On Blood Ties a couple of lovers had this power. After death they would be reincarnated and their old memories would come back once they reached puberty. They would find each other again and they would spend another lifetime as a couple. Things go awry when the woman dies in an accident that also puts the man in a coma for ten years. When he finally reincarnates and gets his memories back he can't find her.

Literature

In the Discworld book Thief Of Time, we find the Abbot of the History Monks does this. He lives a normal life, and remembers each of his previous incarnations, but when he dies, he is immediately reborn somewhere in a village at the foot of the mountain. Lao Tze, on the other hand, has mastered circular aging, so he does not age.

Oral Tradition, Folklore, Myths and Legends

The Phoenix, according to ancient Greek writers. When it reaches a certain age it immolates itself, and a baby phoenix crawls out of the ashes.
Every Dalai Lama is believed to be the reincarnation of the original Dalai Lama, who returns to help his followers rather than Ascending to a Higher Plane of Existence.
Unlike other beings and even other lamas, tulkus, in Buddhism, are able to choose the form of their reincarnation.
Egyptian Mythology: The Egyptian gods are immortal in this way. Although they are depicted as being killed, and aging, in several myths, they always come back to life. The sun god Re dies each and every evening, and is reborn every morning as a child. Though the evidence isn't entirely clear, it appears that all gods were thought to age, die, and resurrect. It is also somewhat unclear, however, whether some of them skip the infancy stage.

Tabletop Games

The Reborn of the Old World of Darkness are effectively human in every way that matters - but every time they die, they're reincarnated as babies, and have to grow up all over again. As they mature, each new incarnation recalls the memories of all their previous incarnations.

Video Games

The fae and arasai in EverQuest II. When they die, either of natural or unnatural causes, a spirit bud is created which regenerates the body until they are ready to be reborn. However, the spirit bud can be destroyed, which leads to the fae or arasai's permanent death.
Dogmeat in Fallout 3, if Broken Steel is installed and a certain perk is taken, gains this form of immortality. Every time he is killed, he is reincarnated in a new body as Dogmeat's Puppy.
From BloodRayne, a Flower Child’s life cycle is a true cycle. They can die and be reborn a seemingly unlimited number of times, provided they aren’t killed before reaching physical maturity. Once they die, a large plant grows from their grave (that plant being whatever type of flower they’re based on). This plant bears a ‘fruit’ which is basically a womb, in which the Flower Child’s new body develops. They are then ‘born’ exactly one year after their death, in the form of a five-year-old child.
The Chao in the Sonic Adventure Series can come back from death this way if raised correctly. In fact, dying and being reborn twice is one of the requirements for a Chao to become a perfectly immortal Chaos Chao.
Michael Roa Valdamjong from Tsukihime invented a magic that allows him to reincarnate in a previously selected infant body if his current one is killed.
Zasalamel from the Soul Series series was originally one of the protectors of Soul Calibur. Against the warnings of one of his fellow tribesmen, Zasalamel discovers the secrets of reincarnation. Over the ages, he regrets this decision, as he realizes that his newfound powers aren't all they're cracked up to be. Thus, he maddeningly seeks the means to end his life for good. After killing himself with Soul Edge and being unable to find the spirit sword he was once tasked with protecting, Zasalamel spends the duration of Soulcalibur III plotting to restore the soul swords to their full power so he can absorb their energies and finally break his cycle of death and rebirth. He succeeds, but sees a vision of the future during his ritual. Amazed by the potential of humanity, Zasalamel, feeling an urge to live again, decides to keep his "curse" and guide mankind toward the vision he glimpsed.
Dracula in the Castlevania series has this in addition to his continual resurrections. After being Killed Off for Real (supposedly) in the Battle of 1999 and his soul sealed away, he was reincarnated to live as a human again, until said human found out he was Dracula and the Sorrow games happened. Interesting in this case because the reincarnation didn't happen immediately. Pieces of Drac's power were incarnated in Graham, Dmitrii, and Dario, who were born at the time he was defeated, but the actual inheritor of his soul, Soma Cruz, was born about sixteen years later.
The historian known as the Child of Miare in Touhou reincarnates in his/her own bloodline every 100~150 years and can recall memories of every incarnation. Hieda no Akyuu is the 9th one. Symposium of Post-mysticism states that some youkai (without specifying which ones) are reborn after their death.

Western Animation

In South Park, a mixture of this and Resurrection Immunity happens to the character of Kenny McCormick. When he dies, his mother instantly has another child, who ages to Kenny's age overnight. Only the new Kenny, who has all the old Kenny's memories, and his parents remember what happened, but the latter assume it's just a drug hallucination. Everyone else just thinks Kenny bailed on them.