How rapidly do supermassive black hole seeds'' grow at early times?

We investigate the physical conditions for the growth of intermediate-mass seed black holes assumed to have formed from remnants of the first generation of massive stars. We follow the collapse of high sigma halos with T-vir > 10(4) K using cosmological, smooth-particle hydrodynamic (SPH) simulations in the standard Lambda CDM model. During the collapse of the parent halo, the seed holes are incorporated through mergers into larger systems and accrete mass from the surrounding gas. We include a self-consistent treatment of star formation, black hole accretion, and associated feedback processes. Even under optimistic assumptions for the seed black hole mass and for efficient merger rates, we find that seed holes in halos M <= 10(10) M-circle dot never reach the conditions for critical Eddington growth. Most of the black hole growth in this regime is determined by the initial mass and the merger rates. Critical accretion rates are reached, albeit only after a significant delay, at the time of collapse (z similar to 7) for 3-4 sigma halos of M similar to 10(11) M-circle dot. Our results imply M-BH = 5 x 10(6) M-circle dot (M-halo/10(11) M-circle dot)(0.78) at the time of collapse. The required conditions of Eddington growth to explain the buildup of supermassive black holes (similar to 10(9) M-circle dot), as implied by Sloan quasars at z > 6, are therefore hard to meet in such a scenario. Without a jump start, these conditions may be only achieved in extremely rare halos with M-halo > 10(13) that collapsed before z similar to 6. The sub-Eddington regime in which black holes accrete at early time implies a small contribution to the reionization by miniquasar but still sufficient to cause appreciable heating of the IGM at z less than or similar to 15-18.