  The large columns of dusty gas enshrouding and fuelling star-formation in
young, massive stellar clusters may render such systems optically thick to
radiation well into the infrared. This raises the prospect that both "direct"
radiation pressure produced by absorption of photons leaving stellar surfaces
and "indirect" radiation pressure from photons absorbed and then re-emitted by
dust grains may be important sources of feedback in such systems. Here we
evaluate this possibility by deriving the conditions under which a spheroidal,
self-gravitating, mixed gas-star cloud can avoid catastrophic disruption by the
combined effects of direct and indirect radiation pressure. We show that
radiation pressure sets a maximum star cluster formation efficiency of
$\epsilon_{\rm max} \sim 0.9$ at a (very large) gas surface density of $\sim
10^5 M_\odot$ pc$^{-2} (Z_\odot/Z) \simeq 20$ g cm$^{-2} (Z_\odot/Z)$, but that
gas clouds above this limit undergo significant radiation-driven expansion
during star formation, leading to a maximum stellar surface density very near
this value for all star clusters. Data on the central surface mass density of
compact stellar systems, while sparse and partly confused by dynamical effects,
are broadly consistent with the existence of a metallicity-dependent
upper-limit comparable to this value. Our results imply that this limit may
preclude the formation of the progenitors of intermediate-mass black holes for
systems with $Z \gtrsim 0.2 Z_\odot$.
