There is, apparently, no way to disable or remove Game Center unless your device is jailbroken. You can disable your Game Center account or, when the totally annoying invitation to join Game Center comes up when you are playing an app, you can click "cancel" numerous times until it leaves you alone. (Your Game Center account, BTW, shows all the game apps you have on your device that are Game Center enabled with your score compared to others, even if you opt out of having a public profile and "friends").
Log into my Game Center account and remove my "nickname" and all personal references, so it's essentially non-functional.
Turn off all notifications for Game Center in Settings
For the one "must-have" game I play frequently, use the numerous (irritating) button tap method to make Game Center quit bugging me when I play it
Remove all games apps on my device that are Game Center enabled with the exception of the one that I really love playing.
Stop downloading any more game apps that are labeled with the Game Center icon (the ones that run through Game Center are clearly marked in the App Store).
Unfortunately those are all temporary. If you purchase something from the App Store, which triggers you to enter your apple password, it will automatically log you into Game Center.