Square has lots of JRPGs out. Chrono Trigger, FF 1-3, the Chaos Rings games, and Final Fantasy Tactics.
Cthulhu Saves The World is great for a more Dragon Quest kind of experience, as is Dragon Fantasy, both of which have surprisingly great writing.
Rogue Touch is my favorite roguelike on the platform. Simple, hard, great. 100 Rogues is good, and Sword of Fargoal is solid.
There are tons of great board/card games, but Summoner Wars has the most RPG elements involved. It's like a card game crossed with a strategy RPG. Best played with friends, of course. If you like this one, you should look into the other Playdek games, they're all amazing and able to justify owning an iOS device on their own.
There are some WRPG attempts and such beyond roguelikes, but I haven't found any I like(The Quest is just okay). Ditto for the 10 or so MMOs I've tried, nothing remotely good there just yet. Somebody else might be able to help you there.
Most games are on iPhone, but for RPGs, you're missing out on the amazing Spiderweb games if you don't get an iPad eventually (or you can get those on PC).