The creation of complex entangled states, resources that enable quantum computation, can be achieved via simple "probabilistic" operations which are individually likely to fail. However, typical proposals exploiting this idea carry a severe overhead in terms of the accumulation of errors. Here, we describe a method that can rapidly generate large entangled states with an error accumulation that depends only logarithmically on the failure probability. We find that the approach may be practical for success rates in the sub-10% range. The assumptions that we make, including parallelism and high connectivity, are appropriate for real systems including those based on measurement-induced entanglement. This result therefore indicates the feasibility of such devices.