Hi starboardmoon, welcome to G2G.
This can be caused by a damaged "Pop3uidl.dbx" file in your Identity's store folder. This file keeps track of which messages on the server have already been downloaded.
A Google search for Outlook keeps dowloading the same messages over and over gets lots of hits. This one from Experts Exchange sounds promising.
To fix the same messages coming from a POP account again and again:
Open your Outlook Express store folder in Windows Explorer.
Click on the file named "Pop3uidl.dbx" with the right mouse button.
Select Delete from the menu