Practically every college and university has summer programs on campus for high school students.  The college gets to use its resources year-around and maybe attract some applicants, and the students get a taste of college without too much work or commitment.  Everyone wins, except possibly the parents who have to write the check.  We won’t list these programs here, because they are everywhere and they are not cheap.  However, a few are free or heavily subsidized:

The Columbia University Science Honors Program (New York City) offers free Saturday classes to high school students.  It’s competitive to get into.  http://www.columbia.edu/cu/shp/

The California State Summer School for Mathematics and Science (COSMOS) has excellent and taxpayer-subsidized 4-week summer programs on the campuses of UC San Diego, UC Davis, UC Irvine and UC Santa Cruz.  Admission is competitive.  Some of our students have done this.
https://cosmos-ucop.ucdavis.edu/
http://www.jacobsschool.ucsd.edu/cosmos/
http://www.cosmos.uci.edu/
http://cosmos.ucsc.edu/

The Telluride Association’s six-week residential Sophomore Seminars and Summer Programs for Juniors are FREE and excellent.  They are conducted at Cornell University, Indiana University, and the University of Michigan. http://www.tellurideassociation.org/programs/high_school_students.html