Australian AMC 2011 - Senior Question 24

Functions $f$, $g$ and $h$ are defined by

$$\begin{aligned}f(x)&=x+2\\g(0)&=f(1)\\g(x)&=f(g(x-1))\quad\text{for }x\ge1\\h(0)&=g(1)\\h(x)&=g(h(x-1))\quad\text{for }x\ge1.\end{aligned}$$

Find $h(4)$.