jmc

Squaring both sides of the given equation, we get $$2+\sqrt{x}=9.$$Then, $\sqrt{x}=9-2=7.$ Squaring again gives $x=49.$

We check our answer by substituting $x=49$ into the given equation: $$\sqrt{2+\sqrt{x}}=\sqrt{2+\sqrt{49}}=\sqrt{2+7}=\sqrt{9}=3.$$Therefore, $x=\boxed{49}$ is the correct solution. (The step of checking the answer is necessary because squaring both sides of an equation sometimes introduces extraneous roots -- solutions which do not actually satisfy the original equation.)