It's actually been awhile since I restarted a Pokemon game. I did so with Red/Blue in order to get all the starters, but that hasn't been necessary in other generations since the third game doesn't mean a fourth starter. If I remember correctly, I restarted four times in order to get my team of Level 100 Mewtwo... never did finish the training, though.
If we're extending it to any game, I've lost count of how many times I restarted Super Mario RPG. Final Fantasy VII was also worth a couple of replays, as was FFIX.