Fast racing, arcade action, and spot-on controls. And, it's free.
Indie critical darling on consoles and PCs, the iOS version controlled well despite using touch. With a controller, it's even better.
Galaxy on Fire 2 HD
This classic space-opera dogfighting and exploration game adapts beautifully to true dual analog-pad controls.
Grand Theft Auto: San Andreas
Rockstar's epic romp through SoCal is back with touched-up graphics, and it has controls that map onto an iOS controller just like on PlayStation or Xbox. Hands-down the best use of an iOS controller to date.
It's on every other platform, and with a physical controller this version is as good as any of them...and less expensive than most.
Sad that Zelda isn't on iOS? Oceanhorn comes close, and you could fool yourself into thinking that's Link when you're using a controller.
Yeah, it's Pac-Man. There's a free lite version on the App Store that's worth a try, and having a real d-pad makes a huge difference.
Riptide GP 2
It's got water racing and lots of tracks, and you can use the analog sticks to do stunts.
Sega's platforming classic, remastered, shows what all retro games have to do: get on the App Store and be controller-friendly ASAP.
An open-world sandbox indie hit that feels like 2D Minecraft and works best when controller and touch are used together.