The vapor-liquid-solid mechanism of nanowire (NW) growth requires the presence of a liquid at one end of the wire; however, Au-catalyzed Ge nanowire growth by chemical vapor deposition can occur at approximately 100 degrees C below the bulk Au-Ge eutectic. In this paper, we investigate deep sub-eutectic stability of liquid Au-Ge catalysts on Ge NWs quantitatively, both theoretically and experimentally. We construct a binary Au-Ge phase diagram that is valid at the nanoscale and show that equilibrium arguments, based on capillarity, are inconsistent with stabilization of Au-Ge liquid at deep sub-eutectic temperatures, similar to those used in Ge NW growth. Hot-stage electron microscopy and X-ray diffraction are used to test the predictions of nanoscale phase equilibria. In addition to Ge supersaturation of the Au-Ge liquid droplet, which has recently been invoked as an explanation for deep sub-eutectic Ge NW growth, we find evidence of a substantial kinetic barrier to Au solidification during cooling below the nanoscale Au-Ge eutectic temperature.