I like games that are:
1. Simple
2. Pretty
3. Dark
4. Story-driven
5. Fun
Not a fan of RPG's or sports games myself, but I do like platformers, silly games, and certain puzzler games. Pulling some out from my Games folder:
Super Dangerous Dungeons: Platformer in the vein of Mario. Great controls for touchscreen.
https://www.youtube.com/watch?v=sq1OvV55Y8c
Ridiculous Fishing: Catch as many fish as you can, then shoot them. Really fun.
https://www.youtube.com/watch?v=vzKcJ8QFbMk
Blek: Gesture-based puzzler.
https://www.youtube.com/watch?v=D0YtH4ftRrw
Monument Valley: Escher-style architecture puzzler with a bit of a story.
https://www.youtube.com/watch?v=wC1jHHF_Wjo
Doodle Jump: Accelerometer tilt-based game. Simple but addictive. Great when waiting in line.
https://www.youtube.com/watch?v=VVWVZJXMklo
Rayman Jungle Run: Gorgeous graphics, fun platformer. Simple run & jump gameplay.
https://www.youtube.com/watch?v=9h4rKk6DhDA
Jetpack Joyride: Simple but fun platformer.
https://www.youtube.com/watch?v=Jzxi8nid9BQ
Monster Dash: Another simple but fun platformer.
https://www.youtube.com/watch?v=hUQcjPjBWXI
Scribblenauts Remix: Simple but infinite game. Beat the level by writing in whatever you want...tornados, rocket ships, swords, ladders, anything.
https://www.youtube.com/watch?v=HuLJZ6JOTHA
Cut the Rope: Use simple physics to cut the rope to feed the baby monster some candy. Classic & fun game.
https://www.youtube.com/watch?v=pFDS2oFmugs
SuperLaser: Kind of like Galaga or 1942, but with prettier graphics. Very straightforward scrolling shooter.
https://www.youtube.com/watch?v=2Ixbut2pJJU
Pako Car Chase Simulator: Isometric car chase game, see how long you can last! Harder than it sounds (not as bad as Flappy Bird, but it's up there).
https://www.youtube.com/watch?v=EkGyYWcyJGE
Two Dots: Like Connect Four on steroids.
https://www.youtube.com/watch?v=uQ9AwBCnZek
Thomas Was Alone: Platformer with a twist - great story & amazing narration.
https://www.youtube.com/watch?v=5K4zjNtQ3y8
Oscura Second Shadow: Dark, moody, atmospheric platformer.
https://www.youtube.com/watch?v=gNuqstmkkvk
Leo's Fortune: Platformer with gorgeous graphics. Absolutely love the visuals. Fun game.
https://www.youtube.com/watch?v=3n0hdulw1zE
Limbo: Another dark, moody, atmospheric platformer. Fantastic artwork, very film noir.
https://www.youtube.com/watch?v=XJih1aOM_ng
World of Goo: Physics-based construction-puzzler.
https://www.youtube.com/watch?v=fvySATGlMV0
Flight Control: Draw lines with your finger to land the planes. Gets harder & more complex as time goes on.
https://www.youtube.com/watch?v=CKYFSZBp4f8
Staying Together: Basically you control two sides of the screen from one side, but each side has different pitfalls. Really fun (and frustrating!) concept.
https://www.youtube.com/watch?v=SY9dNUHeL9w
If I had to pick three to really show off iOS gaming, I'd go with:
1. Leo's Fortune, due to the stunning graphics
2. Momument Valley, because it's a mind trip (also pretty & good lightweight story)
3. Thomas was alone, for the story & narration
Also Limbo, just because I like dark, moody games.