jmc

We use the property that $a\equiv b(\mathrm{mod}m)$ implies $a^c\equiv b^c(\mathrm{mod}m)$.

$333\equiv 3(\mathrm{mod}11)$, therefore $333^{333}\equiv 3^{333}(\mathrm{mod}11)$.

Since $3^5\equiv 1(\mathrm{mod}11)$, we get that $333^{333}\equiv 3^{333}=3^{5\cdot 66+3}=(3^5{)}^{66}\cdot 3^3\equiv 1^{66}\cdot 27\equiv \boxed{5}(\mathrm{mod}11)$.