  We study very-high rate spherically symmetric accretion flows onto a massive
black hole (BH; 10^2 < M_BH < 10^6 Msun) embedded in a dense gas cloud with a
low abundance of metals, performing one-dimensional hydrodynamical simulations
which include multi-frequency radiation transfer and non-equilibrium primordial
chemistry. We find that rapid gas supply from the Bondi radius at a
hyper-Eddington rate can occur without being impeded by radiation feedback when
(n/10^5 cm^-3) > (M_BH/10^4Msun)^{-1}(T/10^4 K)^{3/2}, where n and T are the
density and temperature of ambient gas outside of the Bondi radius. The
resulting accretion rate in this regime is steady, and larger than 3000 times
the Eddington rate. At lower Bondi rates, the accretion is episodic due to
radiative feedback and the average rate is limited below the Eddington rate.
For the hyper-Eddington case, the steady solution consists of two parts: a
radiation-dominated central core, where photon trapping due to electron
scattering is important, and an accreting envelope which follows a Bondi
profile with T~8000 K. When the emergent luminosity is limited below the
Eddington luminosity because of photon trapping, radiation from the central
region does not affect the gas dynamics at larger scales. We apply our result
to the rapid formation of massive BHs in protogalaxies with a virial
temperature of T_vir> 10^4 K. Once a seed BH forms at the center of the galaxy,
it can grow up to a maximum ~10^5 (T_vir/10^4 K) Msun via gas accretion
independent of the initial BH mass. Finally, we discuss possible observational
signatures of rapidly accreting BHs with/without allowance for dust. We suggest
that these systems could explain Lya emitters without X-rays and luminous
infrared sources with hot dust emission, respectively.
