Not to mention the highly selective schools that are part of the 5 college consortium. It's nuts.
Yup-my college kid was accepted to one of the other 5 colleges-wouldn't have made all the nutty requirements at UMass I'm sure. There are a handful of schools that just can't deal with homeschoolers. Yet plenty of highly selective schools with high numbers of applications seem to have no problem. Go figure...
I wouldn't worry about how many schools a kid applies to, unless of course application fees become burdensome or time spent on applications needs to be spent on maintaining senior grades.
My oldest applied to somewhere in the teens (maybe 15?) and the current high schooler will probably end up between 5 and 10. It depends on goals, academic interests, financial aid opportunities and so on.
One school we lived near when my oldest applied put a cap of 6 schools for their seniors. At the 7th school they would refuse to send any information-transcript, teacher recommendations, counselor letter, etc.
I think the most important factor when choosing how many applications to send (beyond budgetary concerns) is to be realistic about admissions chances and keep a good balance between reach schools, likely schools, and safety schools.