Just to say, I received my 2x III today, and had to try stacking. I've always used stacked teleconverters (as long as I've had them), and have had some good results with the 2x II + 1.4x III (+500 f/4 II). I used them plus a Kenko 2x recently again to photograph distant offshore birds (eiders and scoters, common here but they almost never come within half a mile of shore).
These are not good shots but they prove it is possible. The gull washing and man are at 5600mm (500+1.4+2+2+2), and in poor light, manually focused. So sharpness and focus are out of the window - also I had to shoot at ISO 10000 to correctly expose. Still, it proves it's possible. The order was lens-2x II-2xIII-12mm extension tube-1.4x III (I think!). I don't know if order matters with regard to image quality. For scale, the man was nearly 1/4 mile away, on the other side of the lake. Both shots are uncropped, on a 5D3.