Currently developing a mobile/web-based hybrid strategy warfare game, so...yeah, I play it every single day on my iPad =) If any of you mobile based strategy game types would be interested in being part of a live development project (feudal social strategy/warfare), drop me a line.
My favorites have been: Parallel Kingdoms, Turf Wars and Nueroshima Hex. Those are really the only ones that have managed to keep my attention for any real period of time...and the first two only because I'm fascinated with the idea of location based/gps gaming.
I thought I would like a good game on my Android device, but whenever I find one that seems interesting I realized I'd rather play it on my PC. I don't like the silly "casual" games that dominate the Android market, but I understand why they are suited for "phones" (hardly ever use mine as a phone) and larger more in depth games fit better on PC.
The only game I play on the Android right now is Plants vs. Zombies. I have Bard's Tale remake, but again it is one of those games that makes me say "this feels stupid, I should be playing this on PC."
edit- also, why does it seem like 90% of the mobile games were designed by the same artist who combines japanese anime with disney cartoons? If you just look at the games cover pictures they all look they're different versions of Diner Dash. Oh here's diner dash as a f2p city builder, oh Diner Dash as a modern combat RTS, oh Diner Dash JRPG, Diner Dash platformer...etc
Super Hexagon: Game Over. Begin. I've stopped playing this on public transport due to the involuntary cry of OH TITS that I come out with every time I die.
Slay: 90s Windows shareware strategy game ported to iOS; move your men around an island until you've killed off your opponents. Quite boardgamey in a way. (See also the same dev's Mother of all Battles.)
Puzzlejuice: A fast-paced, unholy hybrid of Tetris and Bookworm.
The Last Express: Classic 90s CD-ROM adventure, now playable on your phone. We're living in the FUTURE here people.
Ascension: Chronicle of the Godslayer: Sorta-kinda like Magic: The Gathering but without the card-buying aspect. (Be warned that the app's future is uncertain, as the makers of the physical game are working on their own cross-platform version; the iOS version will definitely get the next expansion but after that, who knows? Still totally worth it IMHO though.)
The Room: Locked-room puzzle game with gorgeous graphics.
Block Breaker 3: One of the nicest Arkanoid clones I've ever played. It's not quite Nervous Brickdown, admittedly, but then again what is? (RIP ARKEDO YOUR WITH THE ANGLES NOW)
Doom 2 RPG and Wolfenstein RPG: A pair of ultra-lite Dungeon Master-style RPGs. John Carmack himself got his hands dirty on these ones, which should tell you everything you need to know about the quality of them (ie. they're AT LEAST as good as Rage)
Eliss: Lovely, lovely multitouch-based game. Join like-coloured blobs together while keeping them away from differently-coloured ones. Much better than I've just made it sound.
Trainyard and Trainyard Express: Lay tracks to guide trains into their like-coloured stations. The free version (Trainyard Express) has a different set of puzzles to the paid-for version so it's worth getting them both.
AaaaaAAaaaAAAaaAAAAaAAAAA!!! (Force = Mass x Acceleration): It's AaaaaAAaaaAAAaaAAAAaAAAAA!!! in the palm of your hand. Works nicely with tilt controls.
Quell and Quell Reflect: Puzzle games where you move a raindrop around a level picking up pearls, your drop travelling continuously until it hits a wall. It's a bit like Amiga game The Power without the shit music YEAH THATS RIGHT I JUST CALLED SNAP SHIT COME AT ME BRO
Gridrunner: Or I could just as easily recommend Caverns of Minos or Super Ox Wars or Goat Up or any of the other Llamatron games. Retro games, but in the style of early 80s coin-op and home computer games rather than the ubiquitous NES style.
Magnetic Shaving Derby: Because who doesn't like games where you guide a razorblade across someone's face using a magnet? I'll tell you who: fools and twats, and I trust you're neither of those.
Spaceteam: HOLY CRAP it's only the best local multiplayer game like EVER. You need one or more friends with iOS devices to play but if you do then oh my days are you in for a TREAT.
Death Ray Manta: The best kind of DRM. Fab psychedelic twin-stick shooter that I'm pretty sure manages to out-Minter Minter.
I've got an imperial fuckton more games on my iPad but those should do for starters.