How do I uninstall an iOS app completely? I deleted and reinstalled a game on iOS. The app still knows my account name.

According to another question there are no leftovers when an app is deleted.

How it is possible that the app still knows my previous login name? How can I make it forget it?

Note: it's a gamecenter app. I also accepted the option to remove it from the gamecenter.

My device runs iOS 6.0.1 and is not jailbroken.
Did the app know your account name when you installed it initially? Also, which game is it?

It is true that when you delete an app, it is completely removed from your device, and all settings are deleted. It is possible however that it still can identify your device and associate it with information its storing on its own servers.