I really like Shining Force, it is a sega genesis game ported to iOS but I've been really enjoying it. If you haven't already played Final Fantasy 1 and 2 on a million other systems that might be an option (though they are overpriced IMO). Phantasy Star 2 is also available, quite cheap - haven't gotten around to playing it though since I've been enjoying Shining Force. Dofus Battles is more of the "stop the wave of mobs" type variety but it is a lot of fun, and there are rpg-like elements such as getting to choose the armor of your warriors, different classes and leveling up. (there is a free version of this you can try out)
Avernum is awesome but it is ipad only.
Edited, Aug 9th 2012 10:31am by Olorinus
When it comes to sitting around not doing anything for long periods of time, only being active for short windows, and marginal changes and sidegrades I'd say FFXI players were the perfect choice for politicians.