At the time of writing, users have reported that iTunes 11.1.5 may persistently ask for a password from time to time. Regardless whether a purchase has been made and even if the iTunes store hasn’t even been accessed at all.
The short answer why this may occur is owing to a bug that ties iTunes with Safari.
For an undetermined reason it appears that iTunes is storing a cookie in Safari’s respository in ~/Library/Cookies, and kicks up the Sign-In Required window whenever it cannot locate the cookie (usually after users have reset Safari or remove all website data) or if the cookie has the wrong value.
Unfortunately, we do not have a viable permanent solution to offer up at time of writing, and identifying the cause is the best we have at the moment so that users can at least mitigate the frequency of iTunes nagging for a sign-in.
For now, the only real solution requires waiting on Apple to fix this architecture.