We present the results of a systematic search for gravitationally lensed continuum Lyman break ''dropouts'' beyond a redshift 7 conducted via very deep imaging through six foreground clusters undertaken with the Hubble and Spitzer space telescopes. The survey has yielded 10 z-band and two J-band dropout candidates to photometric limits of J110~=26.2 AB (5 σ). Taking into account the magnifications afforded by our clusters (1-4 mag), we probe the presence of z>7 sources to unlensed limits of J110~=30 AB, fainter than those charted in the Hubble Ultradeep Field. To verify the fidelity of our candidates we conduct a number of tests for instrumental effects which would lead to spurious detections, and carefully evaluate the likelihood of foreground contamination by considering photometric uncertainties in the dropout signature, the upper limits from stacked IRAC data and the statistics of multiply imaged sources. Overall, we conclude that we can expect about half of our sample of z-band dropouts to be at high redshift. An ambitious infrared spectroscopic campaign undertaken with the NIRSPEC spectrograph at the W. M. Keck Observatory for seven of the most promising candidates failed to detect any Lyα emission highlighting the challenge of making further progress in this field. While the volume density of high-redshift sources will likely remain uncertain until more powerful facilities are available, our data provides the first potentially interesting constraints on the UV luminosity function at z~=7.5 at intrinsically faint limits. We discuss the implications of our results in the context of the hypothesis that the bulk of the reionizing photons in the era 7