Coolness Test (iTunes link) is a well-executed game of skill where you tap a green button as it moves around the screen. You have to avoid the red button, and both move around the screen with each tap. The developers found people were playing for extraordinary lengths of time, and I can see why. You can enter a sort of trance as you focus on the movement of the buttons. It might even improve hand/eye coordination, but all videogames say that, don't they?
Unfortunately Coolness Test, despite the low price of being free, suffers from over 3,000 negative reviews and less than 600 positive ones. Why? If you read the reviews, it's pretty much the deal with any of these one-trick pony apps: you either love it or hate it. I think it's unfortunate the developers called it a "coolness" test, when wrapping it in the obvious skill test it is would have been more apropos. But it is free, and if you're into button-induced trances, it'll do the trick. There's nothing functionally deficient about the game that I saw, so the reviews are simply people who thought it would determine their actual coolness, I guess.
Spin the Ball (iTunes link) is a bit more toy than game but it is what it is. Swipe your finger quickly across the ball on the screen, then use the accelerometer to "balance" the spinning ball on your finger. Just like a Globetrotter, you can keep spinning the ball with your finger, thus infuriating the Generals. Spin the Ball is also free, and seems to summon less rage by reviewers than Coolness Test, but I'd say it's the lesser of these two apps.