There are many great games on mobile... most of them are ports but they are great fun nonetheless.
I love to play point&click-adventures and the iPad is a great device for them:
- Monkey Island 1 + 2 Special Editions
- Broken Sword 1 + 2 + 5
- Back to the Future
- Sam & Max: Season 1
- Tales of Monkey Island
- The Walking Dead: Seasons 1 + 2
- The Wolf Among Us
- 1112 Episodes 1 - 3
- Amerzone
- Atlantis 2 - 5
- Blackwell Legacy/Unbound/Convergence/Decption/Epiphany (soon)
- City of Secrets 1 + 2
- Cognition
- Dracula 1 - 4
- Edna and Harvey
- Fester Mudd
- Gabriel Knight: 20th Anniversary Edition (soon)
- Forever Lost 1 + 2
- Gemini Rue
- Hector 1 - 3
- Hollywood Monsters
- Leisure Suit Larry Reloaded
- Machinarium
- Moebius (soon)
- Myst
- Perils of Man
- RealMyst: Masterpiece Edition
- Riven
- Runaway 1 + 2 + 3
- Sherlock Holmes: The Awakened
- Syberia
- The Inner World
- The Journey Down
- The Last Express
- The Shivah
- Yesterday
Some other puzzle games are also great to play via touchscreen:
- Ace Attorney: Phoenix Wright Trilogy HD
- Blue Toad Murder Files
- Cogs
- Cut the Rope series
- Detective Grimoire
- Device 6
- Fetch
- Ghost Trick: Phantom Detective
- Jacob Jones and the Bigfoot Mystery
- Layton Brothers: Mystery Room
- Save Toshi DX
- Scribblenauts Remix (works great with voice recognition!)
- Superbrothers: Swords & Sorcery EP
- The Cave
- The Room 1 + 2
- Where's My Water/Perry/Mickey
- World of Goo
Other games which work fine with touchscreen controls:
- Star Wars: Knights of the Republic
- XCOM: Enemy Unknown
- Anomaly 1 + 2
- Dark Nebula 1 + 2
- Dungeon Hunter 1 - 3
- FTL: Faster than Light
- Infinity Blade 1 - 3
- Labyrinth 2
- Lili
- Pinball Arcade
- Plants vs. Zombies
- Rayman: Jungle Run + Fiesta Run
- République
- The Bard's Tale (and the included Bard's Tale Trilogy)
And many other games, which work better with a controller, are getting MFi-support: http://mfi4.gamerz.co/topic8.html
(for example: GTA San Andreas, Dead Trigger 2, Galaxy on Fire 2, Jet Set Radio, Lego Star Wars, Limbo, Metal Slug 3, OceanHorn, Prince of Persia, R-Type 1+ 2, Riptide GP 2, Shantae: Risky's Revenge, Sonic games, Tomb Raider)









