I find this even more fascinating since once a letter is written it doesn’t really take any more time to send it to another school. (Thank goodness too between schools and scholarships we did need them to submit their letters several times.)
My kids were at a big HS too, and certain teachers wrote a lot of letters. They may have had an “unadvertised” limit (i.e “I just can’t do one more…gotta start saying no now…”), but as far as I know they didn’t have any “advertised” limit.