The Bouncer for the PS2. It was a fighting game, one of the only ones I like. The story was a little... weird, but you got the option at the beginning of each level to play as one of three characters, and I always thought that was fucking cool.
Dark Cloud, also for the PS2. That was a fucking fantastic game, probably one of my favourites. It had a pretty solid storyline, a unique world, and the settings were just fascinating. The first one, anyways. I never finished the second one, it couldn't capture me the same way the first one did.