When I log into game center for the first time with a new user theres a second game center popup "Welcome to Game Center" after the initial login screen which asks you to select an avatar and a nickname. If you just select "Continue" which is the big blue button in the middle of the screen you will now be taken back to the app (which has been running in the background, not paused like the initial popup).
And if you now go to Settings and log out of game center and restart the app you will see the exact same loop again with a login screen, followed by the "Welcome to Game Center" which asks you yet again to select an avatar and username. I would expect this behaviour to repeat itself if I chose "not now" in the top left corner, but not when pressing Continue.
The pattern stops repeating itself if I ever select an avatar. Even if its just a default avatar equal to the one I had when I just pressed Continue.