I also have this Outlook 2003 problem on multiple PC's in a SBS 2003
environment.

The first time Outlook is opened it works fine.

When you exit Outlook, the icon remains in the taskbar, but it is not
responsive.

Outlook is not listed in the Task Manager as a running application
however Outlook.exe remains resident as a process and new instances of
Outlook attempt to start but will not open until the original
outlook.exe process is ended.

I have not found any info in the KB or elsewhere. Anybody have any ideas?

Some common reasons why Outlook will not shut down completely when you click
either the X or Exit:

1.  PDA synchronization software with your PDA in the cradle.
2.  WinFax Pro (especially 10.02 in Outlook 2003) - contact their support
for an alleged fix.
3.  Franklin-Cover Plan Plus!
4.  COM Add-ins.
5.  Mail reminder add-ins that keep a stub of Outlook open in the background
to check for new mail and fire reminders.
6.  Anti-virus/firewall programs that are set to scan incoming and outgoing
mail.

See if this helps:
http://www.slipstick.com/problems/close.htm