Biodiversity sampling is labor intensive, and a substantial fraction of a biota is often represented by species of very low abundance, which typically remain undetected by biodiversity surveys. Statistical methods are widely used to estimate the asymptotic number of species present, including species not yet detected. Additional sampling is required to detect and identify these species, but richness estimators do not indicate how much sampling effort (additional individuals or samples) would be necessary to reach the asymptote of the species accumulation curve. Here we develop the first statistically rigorous nonparametric method for estimating the minimum number of additional individuals, samples, or sampling area required to detect any arbitrary proportion (including 100%) of the estimated asymptotic species richness. The method uses the Chao1 and Chao2 nonparametric estimators of asymptotic richness, which are based on the frequencies of rare species in the original sampling data. To evaluate the performance of the proposed method, we randomly subsampled individuals or quadrats from two large biodiversity inventories (light trap captures of Lepidoptera in Great Britain and censuses of woody plants on Barro Colorado Island [BCI], Panama). The simulation results suggest that the method performs well but is slightly conservative for small sample sizes. Analyses of the BCI results suggest that the method is robust to nonindependence arising from small‐scale spatial aggregation of species occurrences. When the method was applied to seven published biodiversity data sets, the additional sampling effort necessary to capture all the estimated species ranged from 1.05 to 10.67 times the original sample (median ≈ 2.23). Substantially less effort is needed to detect 90% of the species (0.33–1.10 times the original effort; median ≈ 0.80). An Excel spreadsheet tool is provided for calculating necessary sampling effort for either abundance data or replicated incidence data.