A broad transparency range of its constituent materials and compatibility with standard fabrication processes make germanium-on-silicon (Ge-on-Si) an excellent platform for the realization of mid-infrared photonic circuits. However, the comparatively large Ge waveguide thickness and its moderate refractive index contrast with the Si substrate hinder the implementation of efficient fiber-chip grating couplers. We report for the first time, to the best of our knowledge, a single-etch Ge-on-Si grating coupler with an inversely tapered access stage, operating at a 3.8 μm wavelength. Optimized grating excitation yields a coupling efficiency of -11 dB (7.9%), the highest value reported for a mid-infrared Ge-on-Si grating coupler, with reflectivity below -15 dB (3.2%). The large periodicity of our higher-order grating design substantially relaxes the fabrication constraints. We also demonstrate that a focusing geometry allows a 10-fold reduction in inverse taper length, from 500 to 50 μm.