There are many terrific summer programs that may interest math circle students. Most are at least partly subsidized by grants and/or offer need-based financial aid. A few are free.
The Art of Problem Solving website maintains an extensive list of excellent math summer programs around the country. Here is a link to an excellent panel discussion in which the speakers included leaders of Canada/USA MathCamp, Hampshire College Summer Studies in Mathematics, Math Olympiad Summer Program, PROMYS, Ross, and Stanford University Mathematics Camp.
Albany Area Math Circle members have attended many of these programs, and you may want to talk to one of the students or alumni listed below about their experiences if you are considering applying to one of them.
AwesomeMath Summer Program (Dallas TX/Santa Cruz CA/Ithaca NY)Andrew Ardito (2006), Matthew Babbitt (2008-2009, 2011), Paul Rapoport (2008), Gili Rusak (2012), Alison Miller (staffmember 2007)
Canada/USA Mathcamp (Location moves each summer)Drew Besse (2003-2007), Beth Schaffer (2004), Alison Miller (staff 2009-1010)
Hampshire College Summer Studies in Mathematics (HCSSiM Hampshire College, Amherst, MA) Liz Simon (2008)
Idea Math Zubin Mukerjee (2010), Cecilia Holodak (2011), Gili Rusak (2011)
MathPath (Location moves each summer) Drew Besse (2002), Andrew Ardito (2005)
Math Olympiad Summer Program (Lincoln, Nebraska)
Alison Miller (2000-2004, staffmember in 2007-2012), David Rolnick (2006, 2008), Andrew Ardito (2007), Matthew Babbitt (2010)
Mathematics Triple (Amherst, MA) Cecilia Holodak
MIT Women's Technology Program (Cambridge, MA) Nicole Bieber (2005, staffmember in 2007)
Program in Mathematics for Young Scientists (PROMYS, Boston MA) Manisha Padi (2002), Andrew Ardito (2008-2009;staff member 2010-2011), Zubin Mukerjee (2011-2012)
Research Science Institute (MIT, Cambridge, MA) (Math and science research program) Beth Schaffer (2006), David Rolnick (2007), Matthew Babbitt (2012)
Ross Summer Program in Mathematics Brady Pelkey (2011)
Stanford University Mathematics Camp (SUMaC) Katherine Brainard (2002)
Telluride Association Summer Program (Ithaca, NY; Ann Arbor, MI; Austin, TX) (Humanities program) James Dick (2005), Zagreb Mukerjee (2008), Beatrice Malsky (2010)
The University of Iowa Secondary Student Training Program (SSTP) Noah Rubin (2010)
Young Scholars Program in Discrete Mathematics (Rutgers University, NJ) Peixuan Guo (2009)