+600 Notice that $2013\equiv 11\equiv -2\pmod{13}$.
So $2013^{2013^2}\equiv (-2)^{2013^2}\equiv -2^{2013^2}\pmod{13}$ since $2013^2$ is odd.
Notice that $\gcd(2,13)=1$ and $13$ is prime, so we can invoke fermat's little theorem, i.e. $2^{12}\equiv 1\pmod{13}$.
Notice that $2013\equiv 9\pmod{12}\implies 2013^2\equiv 9^2=81\equiv 9\pmod{12}$.
Hence $-2^{2013^2}\equiv -2^9\pmod{13}$. Note $2^9\equiv 512\equiv 5\pmod{13}$, hence the answer is $-5\equiv \boxed{8}\pmod{13}$.
