RESOLVED for me, anyway (read below)! Ok...this drove me absolutely nuts! My 14 Pro Max was replaced under warranty...a phone I had not updated since it was new a year ago (sorry but if it ain't broke I'm not updating and even if it is, I think 5 times before updating-too many times features are lost in updates!!!). I love how we are told it is a hotmail/Microsoft issue...that is completely wrong...none of my older iphones have problems...they are not on ios17xxx@! I don't like having to install apps that I don't plan to use just to fix problems, but in this case I did. I have 8 different hotmail accounts and every time I would open the mail app on my iPhone it would blow up with re-enter password prompts TWICE for each of them!!! I would hit cancel on each and they would pop up again for each. Ok, here's what worked for me:
(iPhone 14 Pro Max running IOS 17.0.1)
1) SETTINGS --> GENERAL --> TRANSFER OR RESET IPHONE --> RESET --> RESET NETWORK SETTINGS
2) Install EDGE from App Store, add hotmail accounts in EDGE; checking 'stay logged in on this device' for each
3) SETTINGS --> MAIL --> ACCOUNTS --> re enter passwords for each affected account.
4) Tested, all working fine.
5) Delete EDGE app.
All working fine since...why, I am not sure?! (crossing my fingers)
I hope this helps someone else.
Good Luck All!