Semiconducting nanowires (NWs) are important "building blocks" for potential electrical and electromechanical devices. Here, we report on the mechanical properties of supercritical fluid-grown Ge NWs with radii between 20 and 80 nm. An analysis of the bending and tensile stresses during deformation and failure reveals that while the NWs have a Young's modulus comparable to the bulk value, they have an ultimate strength of 15 GPa, which is the maximum theoretical strength of these materials. This exceptional strength is the highest reported for any conventional semiconductor material and demonstrates that these NWs are without defect or flaws that compromise the mechanical properties.