Q: problem with game center

I share the same problem with you. But mine is a newly created account which it was my first time logging in to game center. I encountered the loop just as you did so i tried to login through the app "game center" and settings, i also tried to login the account through different device BUT none of the above worked, it asked my id, terms agreement, nickname again and again.

However, this doesn't appears when i login to another old account.

Mine is a iphone 4 btw, so it seems it appears in all versions.:0

you are not alone and here is another thread with the same problem

Re: unable to create nickname in game center

I have the same problem iOS 10.0.1 on and iphone 6

Solutions from Apple Support. Please refer to comments in Re: unable to create nickname in game center