The limits of areal storage density that is achievable with heat-assisted magnetic recording are unknown. We addressed this central question and investigated the areal density of bit-patterned media. We analyzed the detailed switching behavior of a recording bit under various external conditions, allowing us to compute the bit error rate of a write process (shingled and conventional) for various grain spacings, write head positions, and write temperatures. Hence, we were able to optimize the areal density yielding values beyond 10 Tb/in2. Our model is based on the Landau-Lifshitz-Bloch equation and uses hard magnetic recording grains with a 5-nm diameter and 10-nm height. It assumes a realistic distribution of the Curie temperature of the underlying material, grain size, as well as grain and head position.