  The observational evidence that Super-Massive Black Holes ($M_{\bullet} \sim
10^{9-10} \, \mathrm{M_{\odot}}$) are already in place less than $1 \,
\mathrm{Gyr}$ after the Big Bang poses stringent time constraints on the growth
efficiency of their seeds. Among proposed possibilities, the formation of
massive ($\sim 10^{3-6} \, \mathrm{M_{\odot}}$) seeds and/or the occurrence of
super-Eddington ($\dot{M}>\dot{M}_{Edd}$) accretion episodes may contribute to
the solution of this problem. In this work we analytically and numerically
investigate the accretion flow onto high-redshift ($z \sim 10$) black holes to
understand the physical requirements favoring rapid and efficient growth. Our
model identifies a "feeding-dominated" accretion regime and a
"feedback-limited" one, the latter being characterized by intermittent (duty
cycles ${\cal D} \lesssim 0.5$) and inefficient growth, with recurring outflow
episodes. We find that low-mass seeds ($\lesssim 10^{3-4} \,
\mathrm{M_{\odot}}$) evolve in the feedback-limited regime, while more massive
seeds ($\gtrsim 10^{5-6} \, \mathrm{M_{\odot}}$) grow very rapidly as they are
found in the feeding-dominated regime. In addition to the standard accretion
model with a fixed matter-energy conversion factor ($\epsilon = 0.1$), we have
also explored slim disk models, appropriate for super-Eddington accretion,
where radiation is trapped in the disk and the radiative efficiency is reduced
($\epsilon \lesssim 0.04$), which may ensure a continuous growth with $\dot{M}
\gg \dot{M}_{Edd}$ (up to $\sim 300\dot{M}_{Edd}$ in our simulations). Under
these conditions, outflows play a negligible role and a black hole can accrete
$80\%-100\%$ of the gas mass of the host halo ($\sim 10^7 \,
\mathrm{M_{\odot}}$) in $\sim 10 \, \mathrm{Myr}$, while in feedback-limited
systems we predict that black holes can accrete only up to $\sim 15\%$ of the
available mass.
