We present deep J- and K_s-band photometry of 20 high redshift galaxy clusters between z = 0.8 and 1.5, 19 of which are observed with the MOIRCS instrument on the Subaru telescope. By using near-infrared light as a proxy for stellar mass we find the surprising result that the average stellar mass of Brightest Cluster Galaxies (BCGs) has remained constant at ∼9 × 10¹¹ M_⊙ since z ∼ 1.5. We investigate the effect on this result of differing star formation histories generated by three well-known and independent stellar population codes and find it to be robust for reasonable, physically motivated choices of age and metallicity. By performing Monte Carlo simulations we find that the result is unaffected by any correlation between BCG mass and cluster mass in either the observed or model clusters. The large stellar masses imply that the assemblage of these galaxies took place at the same time as the initial burst of star formation. This result leads us to conclude that dry merging has had little effect on the average stellar mass of BCGs over the last 9-10 Gyr in stark contrast to the predictions of semi-analytic models, based on the hierarchical merging of dark matter halos, which predict a more protracted mass build-up over a Hubble time. However, we discuss that there is potential for reconciliation between observation and theory if there is a significant growth of material in the intracluster light over the same period.