I have updated my iPhone to iOS 6. The problem it keeps asking for my apple id password. I can either click cancel or fill the password, it works just fine but will pop up again. First I tought it was related to one specific app (LINE by NAVER) but not always. I have tried google for answer but had no luck. Any idea? I deleted the LINE app, and reinstalled. I even restored the phone but nothing helped.
I might have a solution to this as I had been getting the same problem and it seems to have stopped now. I think Apple might have changed the minimum requirements for users' Apple ID passwords. My old password consisted of only lower case letters. When I went to sign in with my old password on the webpage - https://appleid.apple.com/cgi-bin/WebObjects/MyAppleId.woa/266/wa/directToSignIn ?wosid=JM0krXkcNbqVwfqQKTtQA0&localang=en_US) it told me that my account had been locked (or similar wording) because I had entered the wrong password too many times. I had been entering my password consisting of just lower case letters.
When I reset my password and was prompted for a new one, I discovered that passwords must have at least 8 characters, including at least one capital and one number. I have changed my password to fit the new requirements. When I received the pop up on my iPhone I entered my new password and haven't been prompted for it again since.
Hope this helps.