  We analyze the large-$n$ behavior of soliton solutions of the integrable
focusing nonlinear Schr\"odinger equation with associated spectral data
consisting of a single pair of conjugate poles of order $2n$. Starting from the
zero background, we generate multiple-pole solitons by $n$-fold application of
Darboux transformations. The resulting functions are encoded in a
Riemann-Hilbert problem using the robust inverse-scattering transform method
recently introduced by Bilman and Miller. For moderate values of $n$ we solve
the Riemann-Hilbert problem exactly. With appropriate scaling, the resulting
plots of exact solutions reveal semiclassical-type behavior, including regions
with high-frequency modulated waves and quiescent regions. We compute the
boundary of the quiescent regions exactly and use the nonlinear
steepest-descent method to prove the asymptotic limit of the solitons is zero
in these regions. Finally, we study the behavior of the solitons in a scaled
neighborhood of the central peak with amplitude proportional to $n$. We prove
that in a local scaling the solitons converge to functions satisfying the
second member of the Painlev\'e-III hierarchy in the sense of Sakka. This
function is a generalization of a function recently identified by Suleimanov in
the context of geometric optics and by Bilman, Ling, and Miller in the context
of rogue wave solutions to the focusing nonlinear Schr\"odinger equation.
