Before you do anything else, close Outlook. You should also note that we'll be modifying system generated files here, so proceed with caution or not at all.
Open up windows explorer window and then paste the following text into the address bar, which should bring you to a folder with a single folder inside with a really long name.
What you'll need to do is just rename that folder to something else (I suggest appending -old onto the end of it so that you could easily rename it back if things go wrong)
Now re-open Outlook, and then type in your password hopefully for the last time, making sure to check the box to remember it. You should now see that the original folder was re-created again.
At this point Outlook should remember your passwords, but you can close Outlook and reopen it to make sure. If you encounter problems with this, you can remove the new folder and then rename the -old folder back to the original name to put everything back the way it was.