'Sign in to iCloud' pop-up won't stop popping up on my iPad
The workable solution (until you get to the Apple store, that is):

Step 1. enter the wrong password and press done until you are presented with the option to reset.

Step 2. press reset and wait until safari is opened.

Step 3. double-click the home button to get the list of opened apps. 

Step 4. kill all of the opened apps including the reset password page.

Your iPhone will work until the next restart.