As with most questions about college admissions, the number of colleges you should apply to depends on individual circumstances. However, as a general rule, I usually advise applying to 7-10 schools. With this number, you can have a few target schools, a few safety schools, and a couple of reach schools. Most applications are pretty involved and require supplemental essays and short answer questions, so I usually don’t suggest attempting to do more than 10. You want to make sure you have plenty of time to devote to each application.
If you are applying to schools in California, as most of my students are, there is a caveat to that number. The UC (University of California) and CSU (California State University) applications let you fill out just one application per institution and send that same application to as many campuses as you would like. Because this is the case, you can apply to as many of the CSU and UC campuses as you like while counting them as just one application each. Keep in mind that you do need to pay an application fee for each campus to which you submit an application, so that may prevent you from sending your application to every UC or CSU campus.