Outlook 2013 keeps bugging me for default mail client

Where and when did you see the pop-up?

When you try to open a .msg file in Windows Explorer?

Either there is no default mail client or the current mail

That's because you have two account in your profile and either the profile is corrupted or the account is it. Try this.

Create a new Outlook profile and configure your account once again, after that, make sure you have selected the new profile as a default one.
See if it works.

Jackie, This window pops up even if I do NOT open Outlook.  It pops up intermittently.

Hemil, how do I get to that window you show? And if I create a new Outlook profile that means adding in all my email addresses again by scratch?  And does that mean that I have to redo all my folders and sub-folders of mail?

You will need to add your outlook account again but with a new Email profile.
You can archive this by surfing on:

1- Control Panel
2- User accounts
3- Mail Microsoft Accounts
4- Windows pop up and click show profile. You should see the current one. Dont delete it, instead add a new one for testing and see of it works.

Also I would recommend to use different profile name to avoid conflict with the current data files.

I dont know with what protocol you have configured your emails but, if you configured your emails via IMAP I dont think you have to recreate your folders. Now, If you have POP3, THEN you need to import all your outlook data back to the new file.

But try that as a testing, sometimes outlook get's buggy when something wrong happens to its data files, usually affects everything.

I have 4 email accounts and all are POP3.

I just ran scanpst.  Let's see if this helped.

My solution worked,