Young Scholars Program
The Young Scholars Summer Program is a four-week day camp for sixth to eleventh graders held at the University of Chicago. The mathematical content switches from year to year -- in odd years, the focus is geometry and probability, and in even years, the focus is on number theory and the theory of field extensions. Students do a significant amount of research. The program is no longer free.
You can find more information here: http://math.uchicago.edu/ysp/