jmc

We see that $f(n)=n+10$ for $n=1,$ 2, 3, $\dots ,$ 9. Then $$\begin{array}{rl}f(10)& =f(5)=15,\\ f(11)& =f(6)=16,\\ f(12)& =f(7)=17,\\ f(13)& =f(8)=18,\\ f(14)& =f(9)=19,\\ f(15)& =f(10)=15,\end{array}$$and so on. At this point, the function becomes periodic, with period 5. Therefore, the maximum value of the function is $\boxed{19}.$