  We examine the stellar mass assembly in galaxy cluster cores using data from
the Cluster Lensing and Supernova survey with Hubble (CLASH). We measure the
growth of brightest cluster galaxy (BCG) stellar mass, the fraction of the
total cluster light which is in the intracluster light (ICL) and the numbers of
mergers that occur in the BCG over the redshift range of the sample,
0.18<z<0.90. We find that BCGs grow in stellar mass by a factor of 1.4 on
average from accretion of their companions, and this growth is reduced to a
factor of 1.2 assuming 50% of the accreted stellar mass becomes ICL, in line
with the predictions of simulations. We find that the ICL shows significant
growth over this same redshift range, growing by a factor of of 4--5 in its
contribution to the total cluster light. This result is in line with our
previous findings for ICL at higher redshifts, however our measured growth is
somewhat steeper than is predicted by simulations of ICL assembly. We find high
mass companions and hence major merging (mergers with objects of masses
$\geq$1/2 of the BCG) to be very rare for our sample. We conclude that minor
mergers (mergers with objects with masses $<$ 1/2 of the BCG) are the dominant
process for stellar mass assembly at low redshifts, with the majority of the
stellar mass from interactions ending up contributing to the ICL rather than
building up the BCG. From a rough estimate of the stellar mass growth of the
ICL we also conclude that the majority of the ICL stars must come from galaxies
which fall from outside of the core of the cluster, as is predicted by
simulations. It appears that the growth of the ICL is the major evolution event
in galaxy cluster cores during the second half of the lifetime of the Universe.
