We present an overview of the asymptotic expansions for high-frequency multiple scattering iterations in the exterior of sound-soft and sound-hard scatterers. These expansions yield wavenumber-dependent estimates on the derivative (of all orders) of the multiple scattering iterations. Such estimates facilitate the design and analysis of Galerkin boundary element methods that achieve frequency-independent approximation accuracy in frequency-independent operation counts. Additionally, we introduce preliminary theoretical developments concerning the accurate approximation of the infinite tail in the Neumann series formulation of multiple scattering problems.