The Canon EF 70-300mm f/4-5.6L IS USM Lens Review has been updated to include Canon EF 1.4x III Extender and Canon EF 2x III Extender compatibility information including ISO 12233 Resolution Chart results. Here are the details:
The Canon 70-300 L is officially NOT compatible with Canon extenders. I've tried mounting Canon extenders to the 70-300 L - and they do not fit. The rear 70-300 L lens element physically hits the front element of the extender.
Then, in a major firmware update announcement for the Canon EOS-1D X, Canon included the 70-300 L and Canon EF 1.4x III Extender combo on a compatibility list.
Intrigued, I did more experimenting. What I learned is that the 70-300 L's rear element retracts into the lens far enough that, at about 250mm, there is enough clearance for Canon extenders to mount.
The available with-1.4x focal length range is about 350-420mm and 500-600mm with the 2x installed. Zooming out wider than the 250mm-or-so zoom ring mark results in a physical bump inside the lens. I'm guessing that it is the rubber around the edge of the extender element contacts the rear 70-300 L lens element or its barrel. I do not recommend mounting this combination due to potential damage the to lens.
I of course felt the need to try out these unsupported combinations. My solution to the damage risk issue was to Gaffer Tape the lens zoom ring to lock it at the 300mm mark while using extenders.
Mounting a 1.4x or 2x extender behind any lens reduces its aperture range by 1 or 2 stops respectively. At review time, only Canon 1-Series bodies can autofocus when using the very-dark f/8 max aperture the 70-300 L lens and 1.4x extender combination yields. Tested 1D X and 1Ds III bodies autofocused this combination.
No bodies can AF with the even-darker f/11 max aperture the 70-300 L and 2x combo yields. While it tries, the tested 1Ds III body cannot lock focus with this combination.
The interesting part of this story starts with the fact that the Canon Extenders do not report their presence when mounted behind the 70-300 L. The reported max aperture incorrectly remains f/5.6.
And what is much more interesting is that the 70-300 L and 1.4x combination's f/8 max aperture combination autofocuses even when mounted to the 60D (and presumably on most/all other Canon EOS bodies). Autofocus speed with the 1.4x extender is very, very slow on all tested bodies, but AF does work - even on the 60D.
Since I'm sure you are wondering what the image quality from these combinations looks like, here is a look at the Canon EF 70-300 L with EF 1.4x III Extender ISO 12233 image quality. Image softening from the 1.4x is very noticeable at f/8 in mid and outer regions of the image circle, though the center of the frame remains reasonably sharp. CA becomes pronounced with increasing amounts visible toward the outer image circle. Stopping down to f/11 results in modestly better image quality.
Here is a look at the Canon EF 70-300 L with EF 2x III Extender ISO 12233 image quality. Wide open, the results with the 2x installed are remarkably close to the performance of the 1.4x III, but more sharpness/contrast degradation is apparent - especially in the center of the frame. The 1.4x III takes a wide lead in the image quality race when stopped down to the with-2x's f/11 max aperture. Diffraction levels the results from both combinations at f/16. Again, the with-2x combination has problems locking autofocus.
While the image quality of the 70-300 L with extenders is not impressive, it can certainly be usable in a pinch (especially in the center of the frame with the 1.4x).