However, in the application window it says "Checking connection to Last.fm....", and the sys tray icon is gone.
If you minimise or close this window, it's not possible show it again because there's no system tray icon. The app appears closed, but is still running in the background and can't be restarted until you close DesktopScrobbler.exe from Task Manager.
Instead, if you choose 'Log In' and go through the auth process once again, it'll work.
Good spot that! I'd added a 'Startup' function that deals with the authorization loop, but for some reason didn't plumb this into the 'Log Out' code, which was basically doing it's own thing!
Windows 7, v. 3.1.6.39
To reproduce: