>>What apps are running when this happens? What're you doing with your machine at the time? <<
I have the following open: Firefox, OSX Calendar,Excel, Preview (printed out some stuff earlier) and that's it. The arrow is under FInder and Dashboard on the Dock, but I haven't had Dashboard open today.
Ahhh, it's sending Calendar alarms. I guess that may trigger it to open so that it can send. Maybe I should just have it create an on screen alarm. I don't use Mail. I use GMail from Firefox. So when Mail opens, it downloads a ton of stuff.