Nathan,
That behavior seems pretty strange.
Three questions for you:
1. If someone logs into the machine with a separate user profile does the crash occur?
2. If you open a blank map in ArcPad, save it and close, does ArcPad crash?
3. If you rename the My ArcPad folder within the My Documents folder, does ArcPad crash when you exit? Rename "My ArcPad" to "My ArcPadOld", or something along those lines.
Windows 7
C:\Users\<UserProfile>\Documents\My ArcPad
Windows XP
C:\Documents and Settings\<UserProfile>\My Documents\My ArcPad
Depending on the results of those tests we may be able to make a better determination as to why the crash is occurring for you.