Why does Outlook keep asking for my password?

Are you in a standalone or cloud (Office 365) environment, or are you connecting to an Exchange server on a local network?

As hypercat stated how are you connected to mail?

If this is exchange you would want to setup NTLM authentication so it uses the cached credentials on the system to authenticate.  Stand-alone you would adjust the account settings to remember UN and Pass.

I have Office 365, not Exchange server.   The same popup happened before I upgraded from Outlook 2007.

First, as Nick Rhode suggested, you should have the user name and password saved in the setup for the account, so that it retains that information and should log in automatically any time a login is required.  It might just be popping up every time it actually goes out to send or receive mail, and saving your credentials would avoid those popups.

Second, it might be caused by intermittent disconnections from the external server. This could be caused by problems at the provider end, either with the company that's hosting your email or with your ISP.  The only way to troubleshoot either one of these would be to contact the service provider.

The UN and password are saved in the setup for the account.   Is there any point in trying to use another email account on another computer, such as Outlook Express to see if the service provider is disconnecting?

It might be the security settings.  In Outlook 2010/2013, you would go to File/Account Settings.  Make sure your account is highlighted and click the Change button.  In that dialog box, click More Settings.  Then click the Security tab.  It's probably set to "Negotiate Authentication."  Try changing it to "Password authentication (NTLM)". Also on this tab make sure that the "Always prompt for logon credentials" dialog box is NOT checked.

There is no security tab under "more settings".

Sorry - my bad.  I was thinking about and looking at a corporate email account that is using Exchange. I'm falling back to the intermittent disconnection issue.  Do you use this account on on this one computer, or do you have other computers or devices downloading this email?  Do you ever connect using the web interface and if so, do you have any logon prompts pop up while online?

I can use it on other computers, but I usually use only one.   I can set up the account under Outlook Express on another computer and see if the disconnection occurs.   If it does not, I will set up the account under Thunderbird to see what happens.   I do not connect using the web interface.   Any other suggestions?

The reason I asked about the web interface is to further test whether the problem is a disconnection issue with the ISP or with the Office 365 service provider.  If you get login prompts from the Outlook Web app while you're actively surfing other websites, then you can pretty safely conclude that the problem is with the Office 365 site itself, not with the ISP.  Testing other email apps will allow you to determine only if it's an Outlook issue as opposed to other email applications.  I doubt that it is being caused by Outlook, so I think it may be a waste of time doing that test, but it's up to you whether you want to give that a try or not.

If the problem also occurred with Outlook 2007, it would seem to suggest that Outlook is the cause.   I will see if the prompt happens with Thunderbird.

I found my own solution by trial and error.