\(\sqrt{2+\sqrt{3+\sqrt{x}}}=3\)

Condition \(x\ge0\)

\(\Leftrightarrow\left(\sqrt{2+\sqrt{3+\sqrt{x}}}\right)^2=3^2\)

\(\Leftrightarrow2+\sqrt{3+\sqrt{x}}=9\)

\(\Leftrightarrow\sqrt{3+\sqrt{x}}=7\)

\(\Leftrightarrow3+\sqrt{x}=49\)

\(\Leftrightarrow\sqrt{x}=46\Rightarrow x=46^2=2116\) (satisfy with condition).

Hence, \(S=\left\{2116\right\}\)