Epsilon-near-zero (ENZ) materials have recently garnered considerable interest for their profound enhancement of optical nonlinearities. In the near-infrared region, the nonlinear response of the ENZ media, notably transparent conducting oxides, has been widely investigated. However, the corresponding research on ENZ materials in the mid-infrared (mid-IR) range remains limited. Herein, we investigate and report the nonlinear optical properties of antimony (Sb)-doped germanium (Ge) around its ENZ wavelength of 4.45 μm. By performing Z-scan measurements at 3.57 μm (away from the ENZ wavelength) and 4.37 μm (near the ENZ wavelength), we report a giant nonlinear refractive index (n₂) in the order of 10⁻¹⁵ m²/W and a complex intensity-dependent absorption that switches from saturable to reverse saturable absorption. Critically, we show that this strong response involves mechanisms beyond the typical intra-band transition, as it also results from phonon-assisted intervalley scattering and multi-photon absorption, which uniquely determine the negative sign of n₂. This work not only provides a foundation for the mid-IR nonlinear optical applications but also reveals the physical mechanism underlying the nonlinear response in ENZ media.