Axion dark matter has the interesting property that it can form overdense substructures known as 'axion stars', whose size and mass depend importantly on the mass and decay constant of the theory. Recently, many authors have considered axion theories with decay constants of the order of the Planck mass, which could collapse to form solar-mass black holes or act as neutron star mimickers. In this talk, I point out that such objects (if they form in the first place) are extremely unstable to relativistic decay processes. By analyzing the dependence of the decay rate on the binding energy and particle mass, I will sketch the parameter space in which axion stars are stable enough to have phenomenological consequences today, but in doing so severely limit the allowed reach of Planck-scale axion stars.