I'm playing Infinity Blade on my iPhone 4 but if I switch to my iPad I have to start the game over, fresh, as if I'd never played before -- it makes me hope Game Center in iOS 5 will sync my game state between devices so I can just pick up any of them and keep playing from where I was. Not just for Infinity Blade but for Angry Birds and Angry Birds HD and tons of other games. All of them, actually.
Currently that sort of sync is left up to the developer. Epic/Chair and Rovio/Chillingo didn't build any sort of sync feature into their games the way Apple did with iBooks, Amazon did with Kindle, Netflix does with their app, or ngmoco's Plus+ network does with their games, so we don't get one in Infinity Blade or Angry Birds.
Jeremy and Leanna have complained about this before and Chad and Alley got into a real fight over it on iPad Live!, but wouldn't it be great if Apple extended Game Center to allow any developer, large or small, to easily add code that would sync game state data for users?
iBooks has your iTunes account, Kindle your Amazon account, Netflix their own account, ngmoco your Plus+ account. I don't want any more accounts. I don't want an Infinity Blade or an Angry Birds accounts. I don't even want a Game Center account, frankly, I'd rather just have my iTunes account cover that as well, but since I have a Game Center account and most games tie into it already, it seems like a great single point of convergence.
It would even leave the door open to gaming on other people's devices. Just log them out of Game Center, log yourself in, launch the game, play, then log out again. Go back to your device, keep playing.
Achievements are nice. Being able to keep them cross-device would be even nicer.