I figured out what was going on, hrdwareguy. I develop software for the Pocket PC, and on my old desktop I just kept my Pocket PC hooked up to my computer all the time because, ironically, I never use it. It syncs with Outlook every 5 minutes automatically, so it keeps Outlook running in the background at all times--I remember hitting CTL-ALT-DEL and seeing "Outlook.exe" running even when it wasn't open. I don't hook it up to my laptop except to debug software, though, so it doesn't do it anymore.
And thanks for the tip, zizdog. I found a registry edit that does the same thing without loading another program, though. Turns out Outlook has system tray functionality built in, there's just no option to turn it on inside the program.
Go figure.
HKEY_CURRENT_USER\Software\Microsoft\Office\10.0\Outlook\Preferences\DWORD MinToTray = 1
Probably only works on Office XP, sorry.