Q: Why do I continue to get asked for a password.

Mail cannot tell the difference between "server did not respond" and "server did not respond because the password is wrong".

Your Mail Server may be doing backups or maintenance, or be offline for a problem. If you know the password is correct, just dismiss the dialog box and try again later.

The problem is "your Mail Server did not respond". You can use:

Mail > Window > Connection doctor

... to see when it does respond.

There is nothing you can do from your Mac that will fix the problem except wait for your mail server, which resides on the Internet, to respond.

If you wish to make the situation more complex, feel free to re-enter and re-enter your password as many times as you like.

Usually after dismissing that dialog box twice it will be quiet for a while.