  The physics of star formation at its extreme, in the nuclei of the densest
and the most massive star clusters in the universe - potential massive black
hole nurseries - has for decades eluded scrutiny. Spectroscopy of these systems
has been scarce, whereas theoretical arguments suggest that radiation pressure
on dust grains somehow inhibits star formation. Here, we harness an accelerated
Monte Carlo radiation transport scheme to report a radiation hydrodynamical
simulation of super star cluster formation in turbulent clouds. We find that
radiation pressure reduces the global star formation efficiency by 30-35%, and
the star formation rate by 15-50%, both relative to a radiation-free control
run. Overall, radiation pressure does not terminate the gas supply for star
formation and the final stellar mass of the most massive cluster is $\sim
1.3\times10^6\,M_\odot$. The limited impact as compared to in idealized
theoretical models is attributed to a radiation-matter anti-correlation in the
supersonically turbulent, gravitationally collapsing medium. In isolated
regions outside massive clusters, where the gas distribution is less disturbed,
radiation pressure is more effective in limiting star formation. The resulting
stellar density at the cluster core is $\ge
10^{8}\,M_{\odot}\,\textrm{pc}^{-3}$, with stellar velocity dispersion $\gtrsim
70\,\text{km}\,\text{s}^{-1}$. We conclude that the super star cluster nucleus
is propitious to the formation of very massive stars via dynamical core
collapse and stellar merging. We speculate that the very massive star may avoid
the claimed catastrophic mass loss by continuing to accrete dense gas
condensing from a gravitationally-confined ionized phase.
