Let $f(x)$ mean that the function $f$ is applied to $x$, and $f^n(x)$ mean $f(f(...f(x)))$, that is $f$ is applied to $x$, $n$ times. Let $g(x)=x+1$ and $h_n(x)=g^n(x)$. What is $h_n^m(0)$?

What is $g^n(x)$ in terms of $x$ and $n,$ and how does that relate to the function $h$?

How about trying to first find an expression $h^2_n(x)$?

By trying values of $m,$ give an expression for $g_n^m(x).$

We have $h_n(x)=g^n(x)=x+n$ since $1$ must be added $n$ times. To compute $h_n^m(0)$ we may first obtain $h_n^m(x),$ and then evaluate it at $x=0.$ Consider that $h_n^2(x)=h_n(h_n(x))=h_n(x+n)=x+2n.$ We can hence inductively see that $h_n^m(x)=x+mn.$ Therefore $h_n^m(0)=mn.$