The whole point of sexual reproduction is that successful organisms pass their (theoretically successful) genes to the offspring instead of aimlessly mutating like bacteria. But when they do, they already lose 50% of their genes in a created egg/sperm, and if there were 3 distinct sexes the gametes would lose 66% of genes, making this variant of natural selection kinda pointless. But, just for fun, what possibilities could there be?
1) Maybe there'd be two distinct kinds of males with advantages in different characteristics, like physical strength and intellect, and conception requires both of them?
2) Bees already have a basically 3-sex system with queens, male drones, and infertile female workers.
3) Jellyfish have an alternating reproductive cycle where one attaches itself to a rock, becomes a polyp and reproduces asexually, and then these created jellyfish breed the regular way. I guess, that's more of a life stage than a separate sex.
4) Some species have trioecy - males, females and hermaphrodites. It's usually seen as transitionary to a regular male-female system and not very evolutionary stable.
5) Maybe there could be 3 sexes in a way that there's female and male who just create gametes and an infertile that bears the fetus? Can't really think of advantages that this system would have.
6) Similar to point 1, but there maybe could exist a YY male along with XY, but that would mean a significantly reduced female population, and that's not evolutionary viable.