Sorry to be pedantic, but are you certain you mean Outlook Express? If you meant Outlook, since Outlook 2000 onwards there is an option to disable the 'new mail' notification icon. The thread at click here refers.
My apologies if you are referring to OE - to my knowledge, there is no equivalent switch for OE.
The only other thing that springs to mind is the 'hide inactive icons' option for the system tray (right-click an empty part of the taskbar, choose Properties, to see this option). However, I have not heard of this option affecting the 'new mail' icon before.