- 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~6-8 galaxies. We report the discovery of 206, 45, and 13 Lyman-break galaxy (LBG) candidates at z~6, z~7, and z~8, 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~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 broad-band filters, which enable very accurate photometric redshift selections. We construct detailed lensing models for all 18 clusters (although some are preliminary) to estimate object magnifications and to identify two new multiply-lensed z >~ 6 candidates. The median magnifications for these 18 clusters are 4, 4, and 5 for the z~6, z~7, and z~8 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 agreement down to ~27 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, in excellent agreement with the lensed expectations and clearly demonstrating the enhanced efficiency of lensing clusters over field surveys. Once again, lensing clusters appear to be a powerful tool in the discovery and study of high-redshift galaxies and allow for the first glimpse of faint galaxies beyond the reach of the deepest HST legacy field surveys, a technique that will continue be exploited with the upcoming ultradeep Hubble Frontier Fields campaign.