At the end of The Gathering Storm, the book implies that the Dragon, Rand al'Thor, can remember all of his past lives. Also, in the Prophecies of the Dragon it says "the many become one". Are there any references to other past lives, besides Lews Therin?
Early on in the series, one of the Forsaken (Ishamael?) told the Dragon that the Dragon hasn't won all of the fights against the Dark One.
At the end of the series, after his revelation at the Dragonmount, he repudiates this with the claim that he's won every battle against the Dark One. We presume that the only way he could have known that with certainty is that he remembers at least some component of each of his incarnations.