  The singular values of a product of $M$ independent Ginibre matrices of size
$N\times N$ form a determinantal point process. Near the soft edge, as both $M$
and $N$ go to infinity in such a way that $M/N\to \alpha$, $\alpha>0$, a
scaling limit emerges. We consider a gap probability for the corresponding
limiting determinantal process, namely, the probability that there are no
particles in the interval $(a,+\infty)$. We derive a Tracy-Widom-like formula
in terms of the unique solution of a certain matrix Riemann-Hilbert problem of
size $2 \times 2$. The right-tail asymptotics for this solution is obtained by
the Deift-Zhou non-linear steepest descent analysis.
