Recommend some mobile games, please...

...but not just any mobile games. I'm frustrated by the fact that all anyone seems to care about making for mobile platforms (dedicated gaming handhelds excluded) are games like Fruit Ninja, Angry Birds, Temple Run, etc. These games are good for a five minute jaunt while you're in line at the post office, but lack any real depth. These games have no story, no interesting mechanics, and perhaps most of all, no draw after the first five minutes are up.

So I'm looking for games that will perhaps give me some of that. Ideally, anything I play will have the following traits:

  • If there's combat, it's turn-based. Touch screens and action RPGs/FPSes tend not to mix very well for me, and that disaster known as Infinity Blade leaves me wary of any knock offs.
  • There should be a story. I'm not expecting anything as riveting as Deus Ex, or as convoluted as Metal Gear Solid, but there should be something beyond a basic premise. I can only work with birds wanting to kill pigs for so long.
  • It has some similarities to things like Persona (or really any SMT game), Final Fantasy, The World Ends With You, Front Mission 4 (yes, I know that's a totally different style of game), or even classic D&D. I know those aren't all in the same vein, and that's intentional.
  • Controls that are actually usable. Like I said, touch screens pose a challenge for control layouts, so ideally, the challenge will have been accepted and the game ends up actually being playable.
With those in mind, can anyone suggest any decent games? So far I'm looking at most of Square's titles, and the Myst games for iOS. I have both Android and IOS devices at my disposal, so don't feel like you have to limit your suggestions to either one. Also, even if there's a game that doesn't fit the guidelines I listed above, if you really feel like it's amazing, please mention it. I'm never opposed to new gaming experiences.
And if you want to discuss the state of mobile gaming, I'd find that an enjoyable read too. Thanks, guys.