I just wanted to say that I am having the same problem with hotmail accounts using IMAP and Outlook 2013 requesting passwords.
They had been all set up using the IMAP server outlook.office365.com and started having this issue mid-January.
During my troubleshooting, I changed a couple to use imap-mail.outlook.com as the IMAP server. Did not make a difference - and it doesn't seem to have any rhyme or reason to why some will ask for the passwords sometimes but other times work just fine.
I also just tried changing one of them to use the outlook.live.com IMAP server. No change.
I currently have 4 hotmail accounts using IMAP (and one gmail account using IMAP). All 4 of the hotmail accounts have this issue.
In addition, I have 4 hotmail accounts using POP. The hotmail account associated with this login uses POP, and I have occasionally had it request the same password sort of thing during this time. The other 3 POP accounts are working just fine at this time.
I will also add that if I just confirm the pop up windows asking for the passwords (which are already filled in with asterisks for the password), it will eventually connect, sometimes after 3-5 tries, and I am able to send and receive with them.
I would be happy to send along information if that is still needed for troubleshooting purposes. I am guessing it is not since the email address was removed for the contact.