Absolutely, if you aren’t strict about “endless”. The Greek theories of reincarnation were incredibly diverse, probably more than Indian, and were delicately integrated with their own philosophy.
There are examples here Reincarnation: An Overview | Psi Encyclopedia which includes references and also other cultures.
Pherecydes of Syros was the first Greek to write about reincarnation, in the 500s BCE, but Greek reincarnation beliefs are more closely related to Pythagoras. Unfortunately, Pythagoras favoured oral instruction and left no writings of his own (those attributed to him are now known to be forgeries). In 440 BCE, Herodotus alleged that Greek reincarnation beliefs had been acquired from Egypt. These teachings affirmed, Herodotus said, that ‘when the body dies, the soul enters into another creature which chances then to be coming to the birth, and when it has gone the round of all the creatures of land and sea and of the air, it enters again into a human body as it comes to the birth; and that it makes this round in a period of three thousand years’. It is not certain that these were Pythagoras’ views, though, and modern scholars doubt that they originated in Egypt. It is not clear that the Egyptians believed in reincarnation; it seems more likely that the elite, at least, subscribed to some form of resurrection instead of reincarnation.
Other Greeks who embraced reincarnation include the poet Pindar (c. 518-c. 438 BCE) and the philosophers Empedocles (c. 492- c. 432 BCE) and Plato (427/428-348/347 BCE). Plato’s ideas are presented in his dialogues, explored through the words of his characters, and are not entirely consistent across his oeuvre. Plato drew on his predecessors but added some thoughts of his own. He likened the soul to a pair of winged horses hitched to a chariot. Some souls lost their wings and became embodied in creatures on Earth. These wingless souls were freed from the body at death to be judged by the gods and might be sent to the underworld to do penance for one thousand years before resuming corporeal life. Most souls required ten such cycles (ten thousand years) before they could regain their wings and return to the gods. Souls on their way to reincarnation were allowed to choose their new bodies, either human or nonhuman animal, but did so on the basis of their temperaments and characters. Before resuming fleshy existence, they were made to drink from the River of Forgetfulness (Lethe) to wipe clean their memories of their previous lives. The notion that the gods judged souls after death and that rebirth occurred after penance had been paid made karma superfluous as an ethical theory for the Greeks.
The problem with endlessness would be their (P.I.E.-like) creation myths implying an absolute beginning, but I believe it’s still quite relevant as even Hinduism has the same creation myth, but they just go back into previous kalpas.
This only glosses over how complicated their theories were. I have also seen from Aristotle that, in order to justify the existence of perception, that we learn what literally all objects are in the previous life, and the reason people disagree over certain perceptions is that some have a less purified soul. Studying philosophy would make their soul more purified, and criminals, slaves, and less virtuous people would have an “impure soul”. The Aristotle / Plato view of pure souls still remains in the west modern day, connected with their lengthy theories of justice and “perfection of man”. This especially inspired medieval Christian theologians and Catholicism (you even see the remnant of ancient monasticism in Catholic priest life).
