$ = not free
Find details, dates, links, etc. by searching institution name on mysummeradvisor.com It will also show other programs at the school besides the ones mentioned in all the posts:
Carnegie Mellon University
Summer Academy for Math and Science (SAMS)
University of Illinois at Urbana-Champaign
Research Apprentice Program (RAP)
Texas Tech University
Clark Scholars Program
Stony Brook University ($)
Research Scholar at GARCIA Polymer Center
National Institutes of Health
Summer Internship Program in Biomedical Research
Massachusetts Institute of Technology
Minority Introduction to Engineering and Sciences (MITES)
Research Science Institute (RSI)
Telluride Association (Humanities)
Telluride Association Summer Program (TASP)
Telluride Association Sophomore Seminar (TASS)
Boston University
Program in Mathematics for Young Scientists (PROMYS)
Stanford University
Stanford University Mathematics Camp (SUMaC) ($)
Center for Talented Youth (CTY) ($)
Questbridge
(several locations and subjects)
Hampshire College ($)
HCSSIM (Math)
Ohio State University ($)
Ross Math Program
University of Nevada, Reno ($)
Davidson THINK Summer Institute
Mathematical Olympiad Summer Program (MOSP)
is an invitation-only based on scoring on the MAA test
http://www.maa.org/math-competitions
LEDA Scholars (recruited from disadvantaged backgrounds)
http://www.ledascholars.org
University of Minnesota (2015 location)
Asian American Journalism J Camp
http://www.aaja.org/jcamp-class-of-2015/
(all ethnicities welcome to apply)