  Consider the product of $m$ independent $n\times n$ random matrices from the
spherical ensemble for $m\ge 1$. The empirical distribution based on the $n$
eigenvalues of the product is called the empirical spectral distribution. Two
recent papers by G\"otze, K\"osters and Tikhomirov (2015) and Zeng (2016)
obtain the limit of the empirical spectral distribution for the product when
$m$ is a fixed integer. In this paper, we investigate the limiting empirical
distribution of scaled eigenvalues for the product of $m$ independent matrices
from the spherical ensemble in the case when $m$ changes with $n$, that is,
$m=m_n$ is an arbitrary sequence of positive integers.
