If you use/install Magic Lantern firmware on your 5DIII, shutter count is available on the camera menu.
Just go to the Magic Lantern menu (accessed via the delete button), across to the debug menu, then scroll down - it's not initially visible, but appears as one of the menu items as you scroll past the bottom of the screen.
ML has a bunch of other useful features for both video and still photography too.
I tend to leave an SD card in the camera with ML on it, so it loads by default, but if I don't want to use it, simply hold the SET button when turning the camera on, and it boots with the standard Canon firmware, so both options are easily accessible.