2002 AMC 12A Problem 19

The graph reaches $6$ at $x = -2$ and $x = 1,$ so $f(f(x)) = 6$ requires $f(x) = -2$ or $f(x) = 1.$ The horizontal line $y = -2$ meets the graph twice, and $y = 1$ meets it four times, giving $2 + 4 = 6$ solutions. Thus, the correct answer is D.