jmc

We have $$\begin{array}{rl}f(f(x))& =f\left(\frac{2x+9}{x-7}\right)\\ & =\frac{2\cdot \frac{2x+9}{x-7}+9}{\frac{2x+9}{x-7}-7}\\ & =\frac{2(2x+9)+9(x-7)}{(2x+9)-7(x-7)}\\ & =\frac{13x-45}{-5x+58}.\end{array}$$Therefore, the equation $f(f(x))=x$ becomes $$13x-45=-5x^2+58x,$$or $$5x^2-45x-45=0.$$By Vieta's formulas, the product of the solutions to this equation is $\frac{-45}{5},$ or $\boxed{-9}.$