A new way to run nested sampling, combined with realistic MCMC proposals to generate new live points, is presented. Nested sampling is run with a fixed number of MCMC steps. Subsequently, snowballing nested sampling extends the run to more and more live points. This stabilizes MCMC proposals over time, and leads to pleasant properties, including that the number of live points and number of MCMC steps do not have to be calibrated, that the evidence and posterior approximation improves as more compute is added and can be diagnosed with convergence diagnostics from the MCMC literature. Snowballing nested sampling converges to a ``perfect'' nested sampling run with infinite number of MCMC steps.

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