You go until you get into a school you like? There is absolutely no way that someone could answer that question. It is far too much on a case by case basis.
So for instance, I have currently in my med school savings account $1600 left after the initial application process and secondaries. I have maybe 2 schools within driving distance and 8 schools at 8+ hours (so flying). So realistically that will afford me mayyybe 3 interviews by flight and 2 by driving. So whichever 5 get to me first and whichever one of those 5 I like the best, I will pick. If it happens to be the first school I interview at and I like them enough, then thats all the interviewing that I will do, I will cancel the rest and pocket the cash so my wife and I dont have to be in the red heh. I did not apply to a single school that I could not see myself going to, so it truly is just whoever gets to me first. If I interview/visit and hate it, then obviously I wont go there, but you get the general idea. If I happen to get lets say 7 interviews and I can only afford 5, then you know I will just cancel the interviews at schools that are lower on my list.
Other people have $10,000 because they saved very well or their parents gave them money. They can obviously afford to keep going to 20 interviews if they like.
I would assume most people are somewhere in between those two options.