Nonlinear structured illumination microscopy (nSIM) is an effective approach for super-resolution wide-fieldfluorescence microscopy with a theoretically unlimited resolution. InnSIM, carefully designed, highly-contrasted illumination patterns are combined with thesaturation of an optical transition to enable sub-diffraction imaging. While the techniqueproved useful for two-dimensional imaging, extending it to three-dimensions is challengingdue to the fading of organicfluorophores under intense cycling conditions. Here, we present acompressed sensing approach that allows 3D sub-diffraction nSIM of cultured cells bysaturatingfluorescence excitation. Exploiting the natural orthogonality of speckles at differentaxial planes, 3D probing of the sample is achieved by a single two-dimensional scan.Fluorescence contrast under saturated excitation is ensured by the inherent high density ofintensity minima associated with optical vortices in polarized speckle patterns. Compressedspeckle microscopy is thus a simple approach that enables 3D super-resolved nSIM imagingwith potentially considerably reduced acquisition time and photobleaching.