jmc

We have that $$\begin{array}{rl}f(f(x))& =f\left(\frac{cx}{2x+3}\right)\\ & =\frac{c\cdot \frac{cx}{2x+3}}{2\cdot \frac{cx}{2x+3}+3}\\ & =\frac{c^{2}x}{2cx+3(2x+3)}\\ & =\frac{c^{2}x}{(2c+6)x+9}.\end{array}$$We want this to reduce to $x,$ so $$\frac{c^{2}x}{(2c+6)x+9}=x.$$Then $c^{2}x=(2c+6)x^2+9x.$ Matching coefficients, we get $2c+6=0$ and $c^2=9.$ Thus, $c=\boxed{-3}.$