Brian said:
Stick an unaddressed message in the Outbox. When you close Outlook it will
tell you you have unsent messages and ask if you want to close.
Since Outlook won't let you send an e-mail (to put it into the Outbox)
with no recipients specified, use File -> Save to put a copy into the
Drafts folder. Exit the new-mail compose window. Drag the item from
the Drafts folder to the Outbox folder. Right-click on the item and
select Mark As Read. You'll get a 15-second prompt before Outlook will
exit to decide if you want to rescind your deliberate or accidental
choice to exit Outlook.