jmc

We must find the distance between each pair of points.

The distance between $(2,2)$ and $(6,2)$ is 4, since these two points have the same $y$-coordinate.

The distance between $(2,2)$ and $(5,6)$ is $\sqrt{(5-2{)}^2+(6-2{)}^2}=\sqrt{9+16}=5$.

The distance between $(5,6)$ and $(6,2)$ is $\sqrt{(6-5{)}^2+(2-6{)}^2}=\sqrt{1+16}=\sqrt{17}$.

Out of 4, 5, and $\sqrt{17}$, 5 is the largest value. Thus, the longest side of the triangle has length $\boxed{5}$.