I agree with zerojay that a fullscreen throbber would be annoying if you wanted to read the screen. The problem with fading is that it might be confused with just the backlight level changing.

Maybe the OS could put a throbber in the title bar? I know that individual apps can do this at the moment, but if an app isn't responding, then I imagine the OS could also draw up in that area. I realise that this would then be ambiguous between whether it's just the individual app that is busy or the whole OS.

Maybe put the busy throbber in the task button area in the top-left, to show that this is a systemwide thing. Clicking on it could cause more information to pop up.