OK, no guessing. Here's what happens.
I have an app called NYT Crosswords. If I'm signed into GC and start NYT iOS/G. displays a notification "welcome back".
I exit and shutdown NYT, the start GC and logout. I exit GC and shut it down.
I start NYT, and now iOS/GC prompts me to login to GC. It will do this every time I start or resume.
What I do is select cancel, switch out of NYT, and switch back to NYT. GC login pops up again, so again I select cancel. I exit, resume, and select cancel for a third time. GC then prompts me if I want to disable using GC. I select OK, and from then on when I start or resume GC, there are no more notifications or login prompts.
Great, but it will come back, and here's how. If you log into the App Store (which you are forced to do to install a free application - brain dead), the logging into the App Store also logs you into Game Center. Exit App Store, fire up NYT, and it's now back at displaying the GC welcome notification.
Not a huge deal, but a stupid design.
Here's what I have done after months of "Cancel x3, then disable" only to have to do it all over again every time I bought an app... Create a different Apple ID just for Game Center.
1. Go to the sign in page for GC.
2. If you have an existing second ID, sign in with that.
3. If you don't already have multiple ID's, click on "Create an Apple ID" and go through the process using a different email from your main Apple ID.
4. Sign out of Game Center.
5. When you are asked to sign in again when you start your game, select cancel 3 times and then disable.
6. Voila. This will not change the default Apple ID you are using for other Apple services, so you can make purchases from iTunes or the App Store with your regular ID and password, and it will not log you into Game Center. I tested it.