jmc

Note that $2004=2^2\cdot 3\cdot 167$. We focus on the large prime $167$ as the powers of $2$ and $3$ in the prime factorization of $2004!$ are going to be much higher. The largest power of $167$ that divides $2004!$ is $\frac{2004}{167}=\boxed{12}$, the answer.