  We consider products of independent square random non-Hermitian matrices.
More precisely, let $n\geq 2$ and let $X_1,\ldots,X_n$ be independent $N\times
N$ random matrices with independent centered entries with variance $N^{-1}$. It
was shown by G\"otze and Tikhomirov and by Soshnikov and O'Rourke that the
limit of the empirical spectral distribution of the product $X_1\cdots X_n$ is
supported in the unit disk. We prove that if the entries of the matrices
$X_1,\ldots,X_n$ satisfy uniform subexponential decay condition, then the
spectral radius of $X_1\cdots X_n$ converges to 1 almost surely as
$N\rightarrow \infty$.
