My problem (just so there is no confusion): Open iTunes on Windows and I'm prompted 3 times for my AppleID password every time I launch iTunes. Once about iTunes Library, another about iCloud, and the last about iTunes again.
Solution: I tried what Joe suggested above and it did not work. I found this page below helpful (even though it was wrote for Mac). Basically, just click Cancel when you're prompted about your iCloud password ("Your iCloud music session has expired") and then manually sign out of your iTunes account (Windows: Account>Sign Out...), close iTunes, Log back in, and it should fix the problem. I was only prompted once for my password after reopening iTunes and then I closed out of it to see if it worked. I was not asked for my password again (it kept me logged in). I hope this solves your issue as well!
http://www.citymac.com/blog/2015/11/14/how-to-fix-annoying-prompts-and-popups-on -mac