You probably have BearShare starting up with Windows when it boots up.
Naturally, since BearStart didn't get the chance to clear the way for you,
BearShare expires.

Since you are obviously not running with Admin privileges the first thing you
should do is log in as an admin and open the User Accounts control panel
applet to change your account type to Computer Administrator.

Now log in to your usual account and change your date back to July 1 2005 manually,

start BearShare manually,

go to the Setup/Options tab and remove the checkmark from
"Launch BearShare on System Startup".

Change your date back to normal.

Open the properties page for the shortcut you use to launch BearShare with
and edit the Target line from bearshare.exe to bearstart.exe.

You can now change your account type back to "Limited User" again if you wish.

That should clear up the expiry problem permanently.