I never had a PS2 growing up, so I missed out on a huge amount Japanese games from multiple genres. The PS2 was kind of a goldmine like that.
Now however, since the DS is being phased out and the 3DS brought in, it has a pretty complete library of these kinds of games, and the games are cheaper now. But I've pretty much only used my DS for Pokemon.
I'm not too big on most RPG's, but I heard good things about Chrono Trigger, Disgaea, Luminious Arc, Solatorobo and the DQ games.
My favourites were Animal Crossing: Wild World, Elite Beat Agents, Chibi Robo and other smaller games.
The World Ends With You
It's an absolute must and easily the most innovative JRPG on the system - even of the generation. People who don't like it that much admit as much, at least.
^ that, though you may or may not want two sets of eyes for it.
I also recommend SMT: Strange Journey as it's ridiculous and awesome.