We constrain the evolution of the brightest cluster galaxy plus intracluster light (BCG+ICL) using an ensemble of 42 galaxy groups and clusters that span redshifts of z = 0.05 − 1.75 and masses of M 500,c = 2 × 10 13 − 10 15 M . Specifically, we measure the relationship between the BCG+ICL stellar mass M and M 500,c at projected radii 10 < r < 100 kpc for three different epochs. At intermediate redshift (z = 0.40), where we have the best data, we find M ∝M 500,c 0.48±0.06 . Fixing the exponent of this power law for all redshifts, we constrain the normalization of this relation to be 2.08 ± 0.21 times higher atz = 0.40 than at high redshift (z = 1.55). We find no change in the relation from intermediate to low redshift (z = 0.10). In other words, for fixed M 500,c , M at 10 < r < 100 kpc increases fromz = 1.55 toz = 0.40 and not significantly thereafter. Theoretical models predict that the physical mass growth of the cluster from z = 1.5 to z = 0 within r 500,c is 1.4×, excluding evolution due to definition of r 500,c . We find that M within the central 100 kpc increases by ∼ 3.8× over the same period. Thus, the growth of M in this central region is more than a factor of two greater than the physical mass growth of the cluster as a whole. Furthermore, the concentration of the BCG+ICL stellar mass, defined by the ratio of stellar mass within 10 kpc to the total stellar mass within 100 kpc, decreases with increasing M 500,c at all z. We interpret this result as evidence for inside-out growth of the BCG+ICL over the past ten Gyrs, with stellar mass assembly occuring at larger radii at later times.