Nonlinear optical (NLO) phenomena such as harmonic generation, Kerr, and Pockels effects are of great technological importance for lasers, frequency converters, modulators, switches, etc. Recently, two-dimensional (2D) materials have drawn significant attention due to their strong and unique NLO properties. Here, we describe an efficient first-principles workflow for calculating the quadratic optical response and apply it to 375 non-centrosymmetric semiconductor monolayers from the Computational 2D Materials Database (C2DB). Sorting the non-resonant nonlinearities with respect to bandgap $E_g$ reveals an upper limit proportional to $E_g^{-4}$, which is neatly explained by two distinct generic models. We identify multiple promising candidates with giant nonlinearities and bandgaps ranging from 0.4 to 5 eV, some of which approach the theoretical upper limit and greatly outperform known materials. Our comprehensive library of ab initio NLO spectra for all 375 monolayers is freely available via the C2DB website. We expect this work to pave the way for highly efficient and compact opto-electronic devices based on 2D materials.
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