3. A $4 \times 4 \times h$ rectangular prism can hold 8 small spheres with a radius of 1 and 1 large sphere with a radius of 2. Then the minimum value of $h$ is $\qquad$ .
3. $2+2 \sqrt{7}$.

As shown in Figure 2, it is easy to see that at this point $h$ reaches its minimum value. Let the center of the large sphere in the middle be $O$, and the centers of the four smaller spheres below be $A$, $B$, $C$, and $D$. Then the distance from point $O$ to the plane $A$, $B$, $C$, and $D$ is $\sqrt{3^{2}-(\sqrt{2})^{2}}=\sqrt{7}$.
Therefore, $h=2+2 \sqrt{7}$.