  [Abridged] We present a robust measurement of the rest-frame UV luminosity
function (LF) and its evolution during the peak epoch of cosmic star formation
at 1<z<3. We use our deep near ultraviolet imaging from WFC3/UVIS on the Hubble
Space Telescope (HST) and existing ACS/WFC and WFC3/IR imaging of three lensing
galaxy clusters, Abell 2744 and MACSJ0717 from the Hubble Frontier Field survey
and Abell 1689. We use photometric redshifts to identify 780 ultra-faint
galaxies with $M_{UV}$<-12.5 AB mag at 1<z<3. From these samples, we identified
5 new, faint, multiply imaged systems in A1689. We compute the rest-frame UV LF
and find the best-fit faint-end slopes of $\alpha=-1.56\pm0.04$,
$\alpha=-1.72\pm0.04$ and $\alpha=-1.94\pm0.06$ at 1.0<z<1.6, 1.6<z<2.2 and
2.2<z<3.0, respectively. Our results demonstrate that the UV LF becomes steeper
from z\sim1.3 to z\sim2.6 with no sign of a turnover down to $M_{UV}=-14$ AB
mag. We further derive the UV LFs using the Lyman break "dropout" selection and
confirm the robustness of our conclusions against different selection
methodologies. Because the sample sizes are so large, and extend to such faint
luminosities, the statistical uncertainties are quite small, and systematic
uncertainties (due to the assumed size distribution, for example), likely
dominate. If we restrict our analysis to galaxies and volumes above > 50%
completeness in order to minimize these systematics, we still find that the
faint-end slope is steep and getting steeper with redshift, though with
slightly shallower (less negative) values ($\alpha=-1.55\pm0.06$,
$-1.69\pm0.07$ and $-1.79\pm0.08$ for $z\sim1.3$, 1.9 and 2.6, respectively).
Finally, we conclude that the faint star-forming galaxies with UV magnitudes of
$-18.5<M_{UV}<-12.5$ covered in this study, produce the majority (55%-60%) of
the unobscured UV luminosity density at 1<z<3.
