\[\boxed{\mathbf{268}.}\]
\[333\ 333^{2} + 777\ 777^{3} =\]
\[= 3^{2} \cdot 111\ 111^{2} +\]
\[+ 7^{3} \cdot 111\ 111^{3} =\]
\[= 111\ 111^{2} \cdot\]
\[\cdot \left( 3^{2} + 7^{3} \cdot 111\ 111 \right) = 3003^{2} \cdot\]
\[\cdot 37^{2} \cdot \left( 3^{2} + 7^{3} \cdot 111\ 111 \right)\ \vdots 37.\ \]