Recently, MacMahon's generalized sum-of-divisor functions were shown to link partitions, quasimodular forms, and q-multiple zeta values. In this paper, we explore many further properties and extensions of these. Firstly, we address a question of Ono by producing infinite families of MacMahon-like functions that approximate the colored partition functions (and indeed other eta quotients). We further explore the MacMahon-like functions and discover new and suggestive arithmetic structure and modular completions.