There are currently issues with Game Center. Cancelling the sign-in prompt 4-5 times in a row (after each time, hit Home, the tap on the game again and repeat) should get rid of the prompt to sign in...at least it used to. It will likely prompt you again the following day. iOS 9 brought along with it, issues regarding Game Center. Apple does not provide a means to disable it fully.
You may want to go into each of your games and if there's an option to disable Game Center within the game (usually under a "settings" or "options" section), you can try that.
Again, these are only temporary fixes if/when they work. We have to wait for Apple to fix it on their end.