None of my games with Game Center support will return data on the local player. the playerID is always null.
None of any other games I play with Game Center will show the "Welcome Back" logo when I start the game.
When I go into the Settings app to try and logout my Game Center account, it just opens a white blank screen. I have to kill the Settings app to recover.
This also happens if I open the Game Center app directly: it's just a blank screen.
I had this problem last week and finally solved it by doing a Reset All Settings but the problem re-appeared a few days later.
I am running iOS 9.1 on a 6 Plus and a iPad mini 2. Both have the same issue which started at the same time, which makes me wonder if this problem has to do with Game Center servers (or my account) rather than with my specific devices.
Some other data points: I've been playing a lot of new games recently; is it possible I've hit some kind of GC limit? Well over 128 games listed in GC. I tried deleting a lot of games on my Mac's Game Center app but that didn't do anything to fix the issue on my iPhone or iPad.
I can still access my Game Center account on my Mac.