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Novel materials, with enhanced light-matter interaction capabilities, play an essential role in achieving the lofty goals of nonlinear optics. Recently, Epsilon-Near-Zero (ENZ) media have emerged as a promising candidate to enable the enhancement of several nonlinear processes including refractive index modulation and harmonic generation. Here, we analyze the optical nonlinearity of ENZ media to clarify the commonalities with other nonlinear media and its unique properties. We focus on transparent conducting oxides (TCOs) as the family of ENZ media with near zero permittivity in the near-infrared (telecom) band. We investigate the instantaneous and delayed nonlinearities. By identifying their common origin from the band nonparabolicity, we show that their relative strength is entirely determined by a ratio of the energy and momentum relaxation (or dephasing) times. Using this framework, we compare ENZ materials against the many promising nonlinear media that have been investigated in literature and show that while ENZ materials do not radically outpace the strength of traditional materials in either the fast or slow nonlinearity, they pack key advantages such as an ideal response time, intrinsic slow light enhancement, and broadband nature in a compact platform making them a valuable tool for ultrafast photonics applications for decades to come.