ISO 12233 resolution chart results for extenders have been added to the Canon EF 28-300mm f/3.5-5.6L IS USM Lens Review.
According to Canon, the EF 28-300mm f/3.5-5.6L IS USM Lens is not compatible with extenders. Not fully compatible is correct, but ... extenders can indeed be used with this lens as proven by the 420mm and 600mm ISO 12233 chart results in this review.
If the 28-300 L is zoomed out to 50mm or longer, the rear element retracts into the lens far enough for the Canon 1.4x and 2x extenders to physically mount onto the lens. Once installed, zooming in to 50mm will result in physical contact between the front of the extender (likely the rubber ring surrounding the glass lens element) and the rear element of the lens. You obviously want to avoid this.
You will also likely want to avoid the image quality that the Canon EF 2x III Extender combo delivers at 600mm. The results are not pretty. At 600mm, this is an f/11 max aperture lens that will not AF even on a 1-Series body (tried it - AF didn't work). Better, but still not impressive are the results with the Canon EF 1.4x III Extender at 420mm where this is an f/8 max aperture lens. Autofocusing with the 1.4x attached is very, very slow and very frequently fails completely on the EOS 1Ds III being used for testing.
The extender and lens combo's focal length reported properly in EXIF, but the aperture did not.