TibiaME is the world's first Mobile Massively Multiplayer Online Role-Playing Game, or MMMORPG. It allows you to explore a vast world and complete tasks to increase the capabilities of your character.
It does require either WiFi access or an unlimited data plan from your provider, since it IS online. But it's a ton of fun, that's for sure.
This is the free download version, there's also the Gold edition, that's a one-time fee of $5 (not bad) and gives you a few more features. You can also subscribe for $3/month to get a better character.