I have been a Microsoft Office user on the Mac for nearly 3 years now at my workplace and for the last 6-8 months, I have been plagued by repeated password prompts on Outlook 2016 for my Work Exchange account. The sad part is that it just refuses to connect even after putting in the credentials. It seems to happen mostly when connected on the office network and the older Outlook 2011 app does not suffer from this problem. It seems to be a pretty common issue given the number of threads on the Microsoft support forums.
The issue was not present in the initial releases of the Outlook 2016 app and in the last few releases it has started prompting me with the Office 365 sign in page. The issue seems to stem from a conflict with Keychain access on the recent Mac versions, and the only reliable solution I have found is to delete the Exchange related keychain entries as suggested in this thread.
Open up Mac Keychain tool.
Remove *ALL* (delete) references to:
“Microsoft Office Identities Cache 2”
“@<something>” that has a type of “MicrosoftOffice15**”
“Microsoft Office Identities Settings 2”
Update: Received this tweet from the Outlook for Mac Principal Lead Programmer, Alessio, promising a fix in the 15.31 update in Feb 2017. So, keeping my fingers crossed.