We study the $\ell^\infty \to \ell^\infty$ operator norm of products of independent random matrices with independent and identically distributed entries. For $n$-by-$n$ matrices whose entries are centered, have unit variance, and have a finite moment of order $4α$ for some $α> 1$, we find that the operator norm of the product of $p$ matrices behaves asymptotically like $n^{\frac {p+1}{2}}\sqrt{2/π}$. The case of products of possibly non-square matrices with possibly non-centered entries is also covered.