High-temperature series expansions for thermodynamic functions of random-anisotropy-axis models in the limit of infinite anisotropy are presented, for several choices of the number of spin components, m. In three spatial dimensions there is a divergence of the magnetic susceptibility χM for m=2. We find Tc/J=1.78±0.01 on the simple cubic lattice, and on the face-centered cubic lattice, we find Tc/J=4.29±0.01. There is no divergence of χM at finite temperature for m≥3 on either lattice. We also give results for simple hypercubic lattices.