Title explains it. Even when I am asking this question on google chrome, Writing assistant is still on the screen.
Palettes have the setting
WindowFloating -> True which causes them to float above other windows inside Mathematica, but these should not persist when focus is given to another application. However in my experience palettes occasionally become momentarily "stuck" on top but simply switching focus an extra time or two usually fixes it. If this is unsuccessful restarting Mathematica has never failed to fix it. If you have a persistent always-on-top palette that survives a restart and focus changes please give the details of your version and operating system.