Mass accretion by black holes (BHs) is typically capped at the Eddington rate, when
radiation's push balances gravity's pull. However, even exponential growth at the
Eddington-limited e-folding time t_E ~ few × 0.01 billion years is too slow to grow
stellar-mass BH seeds into the supermassive luminous quasars that are observed when
the universe is 1 billion years old. We propose a dynamical mechanism that can trigger
supra-exponential accretion in the early universe, when a BH seed is bound in a star cluster
fed by the ubiquitous dense cold gas flows. The high gas opacity traps the accretion
radiation, while the low-mass BH's random motions suppress the formation of a slowly
draining accretion disk. Supra-exponential growth can thus explain the puzzling emergence
of supermassive BHs that power luminous quasars so soon after the Big Bang.