Password method changed on nexus 4
The prompt for the password is probably caused by one of those apps. Try to remember the password you may have used for them.
 Or else please provide a list of the apps you've tried. (This can be achieved by going to the Google Play website and clicking on my apps. 
As a last resort you can restore your phone to it's factory settings by entering the recovery. 
To boot into the recovery press and hold power and volume up