I have 15, is that too much? What’s too low?
All you really need is one place that you know will admit you because your stats guarantee you admission there, that you know you can pay for with no aid other than aid that your stats guarantee you and/or federal and/or state aid that you qualify for, that offers the major you think you want, and that you like enough to attend.
If you want more options than just that one absolutely safe college/university, then the number of places on your list will depend on other factors such as a need to be able to compare financial aid packages (run the Net Price Calculator at each website first to find out if there is a chance of the place being affordable) and where your stats make it reasonable for you to be admitted.
Some students only apply to one. Some apply to 4. Some apply to 21.
Personally, I think if you do your homework ahead of time (really understanding what each college has to offer - academic, social, financial, etc) most students should be able to get the number of applications to 8 or fewer. Fewer applications means you can devote more time to each and to making sure those schools know that you interested (by visiting, interviewing, contacting with questions, etc).
Many students will apply to more, especially if they are hoping for admittance to one of several very selective schools where the likelihood of admittance is low for everyone.
If you’ll need financial aid or merit scholarships in order to be able to afford to go to college , then you should cast a wide net.
5-8 seems like a fairly typical range, but 10 seems like a reasonable maximum.
The amount of places you apply to should probably correlate with the uncertainty of admission and finances to the schools you’re applying to.
I believe that a FAFSA form will accommodate 10 (although I could be mistaken). Using that number as a maximum, think about applying to 2 safeties, 2-3 reaches, and 5-6 matches; this should give you a broad range. Unless money is no object, and you don’t mind paying all those application fees.