I'm not sure that there's an official limit. I've had 9; 5 on my left and 4 on my right and just had my left knee reconstructed and a tibial tubercle osteotomy on Monday. This was not an arthroscopy but an open knee operation.
Have you searched the net to see if you could get more information? Good luck and if you find an answer please let us know, I'm sure there are many people here who would be curious to know the anwer.
keeping in mind that there are many other things that can be done by arthroscopy besides shaving. I've had Lateral Releases done on both knees arthroscopically, micro-fractures on both knees arthroscopically along with the general cartiledge repairs.
I have had about 15 knee surgeries - about 12 of them arthroscopic and about 10 on one knee (rt). I go in every 2-3 years for an arthroscopy to "clean out" some of the damage. It keeps my knee functioning well. My first knee surgery was at 14 and I'm now 42. I see my orthopedic knee surgeon next week and I'm betting I'm up for another arthroscopy before the end of the year.