Quote from Sedia;1201673:
Puzzle and Dragons. The english version is in the US store and it's like bejeweled but with fighting monsters and stuff.
This is what's been dominating the Japanese smartphone app scene so far. It's very pay2win though, and honestly I'm too into it to stop. Don't play it if you know what's best for you, but I can't stop you from trying.
They've ported a lot of the Final Fantasy games(they've now ported all of the NES/SNES titles, although maybe not in their best form) and The World Ends With You into the iOS platform, so you can try a hand at those if your wallet's full(or if you just jailbreak).
Battle for Wesnoth, a pretty well-known free TBS, is also on the app store for 3.99 if you're into that. Kemco has several generic RPGs that they release weekly for some odd reason.
For those Actiony or those quick games on the go I'd recommend Badland, Punch Quest, Random Heroes, League of Evil, and Tiny Wings. There's also some ported Metal Slug games but touch controls are pretty gimmicky on a run-n-gun.
Most of the smartphone scene is mostly pay2win; there's a few gems in the rough, but they're hard to find.