Six? I count three. One of which is not strictly necessary to get around in in three dimensions. If you can pitch and roll, which are already three dimensional maneuvers in a real sense, you can get along fine. Now, I agree that yaw is a pretty useful function.
Getting out of the shuttlebay already calls for an additional translation - if you can't move straight up and down, you are going to scrape nasty grooves in the floor of the bay every time you attempt takeoff or landing with a craft lacking wheels!
The same with the helicopter-style landings a shuttle is often asked to perform. Under the pull of gravity, you could slide sideways or back/forth by tilting and applying upward thrust, the way a chopper does it, but that won't work if you want to dock in zero gee - you need full three axes of translation in addition to full control over three angles of rotation.