I have an EF-EOS-M speedbooster for my M50. I just got a great deal on a FD 135mm 2.5 lens, and was wondering if I can put it together with an FD-EF adapter and the speedbooster. There doesn't seem to be much flange distance on the FD adapter.
Does your FD-to-EF adapter have a glass element? If so, then yes, you can do it, but the results will be fairly poor, IMO.
If your FD-to-EF adapter doesn't have a glass element, then your proposed combination will cause you to not be able to focus to infinity. This is because the flange focal distance of the FD mount is 42 mm is less than the flange distance of the EF mount's 44 mm. Any FD-to-EF adapter that doesn't have corrective optics is already 2 mm too much, in addition to the thickness of the adapter itself.
But as I said, if the FD-to-EF adapter does have a corrective optical element, then the back-focus and flange focal distances will all be correct. But the corrective element (which usually isn't as high quality as the elements in even most cheap lenses by today's standards), in addition to the optical element in the speedbooster, means you're just stacking a bunch of less-than-highly optimized optical elements together. It will work, and generate images that might be fun to say you've done it, but you'd be much better off just getting a FD-to-EOSM adapter to begin with.
You probably can do it, but stacking adapters increases the possibility of manufacturing inaccuracy causing focus issues (either focusing past infinity, if the adapter combination's too thin, or not focusing to infinity if the combination's too thick.
The main thing to keep in mind, however, is that you will lose all the electronic communication with a speedbooster. The FD/FL mount doesn't have the electronic communication the EF mount does (EF stands for electronic focus, btw), and any FD → EOS adapter won't be able to pass-through electronic communication signals. You'll be limited to manual focus, and manual aperture (i.e., you have to use the aperture ring on the lens). There won't be lens EXIF information, and you'll have to do stop-down metering. And without camera control of the aperture, you're limited to only the M and Av shooting modes.
The only way to know what will happen is to try it. However, you will likely obtain much better results with a plain FD-to-EOS-M adapter or an FD-to-EOS-M focal reducer.
For an FD-EF adapter without optics: My experience with a generic focal reducer (not Speedbooster) is that it can correct for small focus misalignments. However, the difference in flange focal distance + adapter thickness is likely far beyond what the speedbooster could correct. You would still be able to use the lens for close-up work.
For an FD-EF adapter with corrective optics: The corrective optics often behave like a 1.4x teleconverter. The focal reducer will basically cancel this out so that the final focal length will be what you started with (1.4 * 0.72 = 1.01), so you might as well use a plain FD-to-EOS-M adapter.
As far as image quality is concerned, adding glass does not necessarily decrease image quality. Otherwise, you would obtain optimal image quality with a 0-element lens, in the absurd case. Speedboosters have been reported to improve some aspects of image quality.