  We utilize 16 band Hubble Space Telescope (HST) observations of 18 lensing
clusters obtained as part of the Cluster Lensing And Supernova survey with
Hubble (CLASH) Multi-Cycle Treasury program to search for $z\sim6-8$ galaxies.
We report the discovery of 204, 45, and 13 Lyman-break galaxy candidates at
$z\sim6$, $z\sim7$, and $z\sim8$, respectively, identified from purely
photometric redshift selections. This large sample, representing nearly an
order of magnitude increase in the number of magnified star-forming galaxies at
$z\sim 6-8$ presented to date, is unique in that we have observations in four
WFC3/UVIS UV, seven ACS/WFC optical, and all five WFC3/IR broadband filters,
which enable very accurate photometric redshift selections. We construct
detailed lensing models for 17 of the 18 clusters to estimate object
magnifications and to identify two new multiply lensed $z \gtrsim 6$
candidates. The median magnifications over the 17 clusters are 4, 4, and 5 for
the $z\sim6$, $z\sim7$, and $z\sim8$ samples, respectively, over an average
area of 4.5 arcmin$^2$ per cluster. We compare our observed number counts with
expectations based on convolving "blank" field UV luminosity functions through
our cluster lens models and find rough agreement down to $\sim27$ mag, where we
begin to suffer significant incompleteness. In all three redshift bins, we find
a higher number density at brighter observed magnitudes than the field
predictions, empirically demonstrating for the first time the enhanced
efficiency of lensing clusters over field surveys. Our number counts also are
in general agreement with the lensed expectations from the cluster models,
especially at $z\sim6$, where we have the best statistics.
