jmc

Since $12!=12\cdot 11!$, we can examine the sum better by factoring $11!$ out of both parts: $$11!+12!=11!+12\cdot 11!=11!(1+12)=11!\cdot 13.$$Since no prime greater than 11 divides $11!$, $\boxed{13}$ is the largest prime factor of $11!+12!$.