Disagree with Lost Planet, despite not having played the series past 1. 2 sold a little more than 1 did, and it has its own set of fans who enjoy it for the more Monster Hunter style gameplay.
If we talk two game series, then there are a million examples. But sticking to ones that have three or more:
Commando: Capcom's down shooter series. I think it's generally agreed that the second one, Mercs, is the best. The original was a million seller and probably pretty good for its time, but it seems rather dated now. The third is some Western outsourced game that I don't think anyone cares about.
Saturday Night Slam Masters: Saturday Night Slam Masters II is technically the third game in the series, although the second was more of Street Fighter style expansion with single player removed for some reason. Anyway. Slam Masters 2 switches from a wrestling game to a fighting game, which is certainly weird. But the fighting's improved, you can finish people off with your special moves, there are now more special moves and they're a bigger part of the gameplay, etc. Plus, the previous two games has some annoying flaws. Slam Masters didn't have a grab button so grappling was more luck-based, whereas Muscle Bomber Duo was tag team only and almost impossible to win if you CPU partner was pinned.
Konami's Teenage Mutant Ninja Turtles: It's generally agreed that Turtles in Time is the best, right? Maybe some argument between the SNES and the arcade version.
World Heroes: Four games and yet Perfect's the only one people care about.
LP3 has its own warriors too. Written by the guy behind Anachronox, the story is supposed to be really great.