A $6\times6\times h$ rectangular box contains a sphere of radius $2$ and eight smaller spheres of radius $1$. Each small sphere is tangent to three faces of the box, and the large sphere is tangent to each of the small spheres. What is $h$?

1. Place the box as $[0,6]\times[0,6]\times[0,h]$. The small spheres are centered at $(1\text{ or }5,\ 1\text{ or }5,\ 1\text{ or }h-1)$ and, by symmetry, the large sphere is centered at $(3,3,\frac h2)$.
2. Tangency to the small sphere centered at $(1,1,1)$: the distance between centers is $2+1=3$, so $2^2+2^2+\left(\frac h2-1\right)^2=9$.
3. Hence $\left(\frac h2-1\right)^2=1$, so $\frac h2-1=1$ and $h=\boxed{4}$.