Game Center, can't sign up IOS 6.1.6 Ipod touch 4th gen
Just had this problem with an iPod running iOS 6.1.6.  Here are the steps I followed to fix it.


Log into Game Center on a friend's device (must be iOS 7 or greater).
Accept the terms and conditions and create Game Center profile.  Once profile is created, log out of Game Center.
On the iOS 6 device, sign out of your Apple account used by your device and then sign back in.
Launch Game Center and sign in.


Basically, iOS 6.1.6 is having issues when new Game Center users need to create their profile for the first time.