jmc

We note that the domain and the range of $f(x)$ are the same set, $\{1,2,3,4,5,6,7,8,9\}$, and each point in the range is $f(x)$ for exactly one $x$ in the domain. (Thus, $f(x)$ can be said to $permute$ the integers 1 through 9.)

Since the list $f(1),f(2),f(3),\dots ,f(9)$ contains each number from 1 through 9 exactly once, the same must also be true when we apply $f$ again to each number in this list. Thus, the list $f(f(1)),f(f(2)),f(f(3)),\dots ,f(f(9))$ also contains each number from 1 through 9 exactly once, and $$f(f(1))+f(f(2))+f(f(3))+\cdots +f(f(9))=1+2+3+\cdots +9=\boxed{45}.$$