  We study the fundamental rogue wave solutions of the focusing nonlinear
Schr\"odinger equation in the limit of large order. Using a recently-proposed
Riemann-Hilbert representation of the rogue wave solution of arbitrary order
$k$, we establish the existence of a limiting profile of the rogue wave in the
large-$k$ limit when the solution is viewed in appropriate rescaled variables
capturing the near-field region where the solution has the largest amplitude.
The limiting profile is a new particular solution of the focusing nonlinear
Schr\"odinger equation in the rescaled variables --- the rogue wave of infinite
order --- which also satisfies ordinary differential equations with respect to
space and time. The spatial differential equations are identified with certain
members of the Painlev\'e-III hierarchy. We compute the far-field asymptotic
behavior of the near-field limit solution and compare the asymptotic formul\ae\
with the exact solution with the help of numerical methods for solving
Riemann-Hilbert problems. In a certain transitional region for the asymptotics
the near field limit function is described by a specific globally-defined
tritronqu\'ee solution of the Painlev\'e-II equation. These properties lead us
to regard the rogue wave of infinite order as a new special function.
