I had that as well when I did the iOS 8 beta install the other day and tried everything and anything and finally resorted to restoring to the beta and setting up my iPhone as a new device and then sync back everything.
BUT!
One thing I had to do differently. My iCloud account is a @me.com account. When entering that into Settings - iCloud I had to write @icloud.com instead. That and setting up as a new device first, finally made the annoying login-popup go away for good.