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Are you looking for educational? I don't have any of those to recommend. Your request is also a bit... broad, but here goes!

Angry birds (there are over half a dozen)
Cut the rope (currently two of them, soon to be three)
Tiny wings
Jetpack joyride
Anything by Zynga (lots of word games)
Ticket to ride (most people don't know about this game, but its a whole lot of fun)
Sonic (I think there are 2-3 ports and two originals)

The following developers also make some great stuff:
Konami
EA
Sega
Capcom
 
My favorites:

1) Glow Hockey 2
2) Street Fighter II Collection
3) Tetris
4) Sonic CD

Those are my must have games.
 
Are there free games where you can play with two iPhones against each other?
What about cooperative games?
 
Are there free games where you can play with two iPhones against each other?
What about cooperative games?

None that I am aware of. There isn't a whole lot free, but most everything good is a buck or two. If you have both devices on the same iTunes account you only need to buy the game once.
 
I like sports games like:

NFL Quarterback Flick
StarGolf
Homerun Battle 2
Asphalt 7

But one of my favorite games that I have ever played is The Room. This game is mesmerizing and addictive. The only bad thing is that it will take you a day or two to beat it and then you have to wait until this summer for it to be updated again.
 
None that I am aware of. There isn't a whole lot free, but most everything good is a buck or two. If you have both devices on the same iTunes account you only need to buy the game once.

Could you recommend the ones that are a buck or two?
 
I would recommend you few games and all are my favorite...

Need for speed – undercover
Angry birds
Highway Riders
Temple Run 2

All these are very nice games.
 
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