Chiron

Every other centaur descends from Ixion, a man who tried to assault Hera and was tricked into fathering a race on a cloud. Chiron does not. He is the son of Kronos, who took the form of a horse when Rhea caught him with the nymph Philyra — which makes Chiron a half-brother of Zeus, immortal by birth, and no relation at all to the centaurs he is usually counted among. The myths are careful about this, and it explains everything else about him.

He kept a cave on Mount Pelion and ran what amounts to the only school in Greek myth. Asclepius learned medicine there and became the god of it. Jason was raised there. Achilles was brought to him as a child and fed on the marrow of lions. Actaeon, Aristaeus and by some accounts Heracles and Ajax all passed through. Chiron taught medicine, hunting, music and prophecy, and produced more famous graduates than any god managed.

His end is the cruellest thing in the cycle and it was an accident. Heracles, fighting off other centaurs, loosed an arrow dipped in the Hydra's blood and struck Chiron by mistake. The wound could not kill him, because he could not die, and it could not heal, because nothing heals Hydra venom. He was left with pain that had no end available to it.

So he traded. Prometheus was chained and needed an immortal willing to take his place; Chiron gave up his deathlessness in exchange, and was allowed to die. Zeus set him in the sky afterwards. It is the only case in Greek myth of a god choosing mortality, and he chose it to be released rather than to become human.