We use the recently-proposed robust inverse scattering transform for arbitrary spectral singularities and study fundamental rogue wave solutions of the focusing nonlinear Schr\"{o}dinger equation in the limit of large-order. We establish the existence of a limiting profile of the rogue wave of order $k$ 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\"{o}dinger equation in the rescaled variables. This profile — the rogue wave of infinite order — also satisfies ordinary differential equations with respect to space and time, and the spatial differential equations are identified with certain members of the Painlev\'{e}-III hierarchy. We obtain the far-field asymptotic behavior of this near-field solution and also compute it numerically. The same solution is also found to describe near-field asymptotics in an analogous large-order limit of solitons on a zero-background. These properties lead us regard the rogue wave of infinite order as a new special function.