We present the results of a survey for super Lyman limit systems (SLLS; defined to be absorbers with 19.0 <= log(NHI) <= 20.3 cm^-2) from a large sample of high resolution spectra acquired using the Keck and Magellan telescopes. Specifically, we present 47 new SLLS from 113 QSO sightlines. We focus on the neutral hydrogen frequency distribution f(N,X) of the SLLS and its moments, and compare these results with the Lyman-alpha forest and the damped Lyman alpha systems (DLA; absorbers with log(NHI) >= 20.3 cm^-2). We find that that f(N,X) of the SLLS can be reasonably described with a power-law of index alpha = -1.43^{+0.15}_{-0.16} or alpha = -1.19^{+0.20}_{-0.21} depending on whether we set the lower N(HI) bound for the analysis at 10^{19.0} cm^-2 or 10^{19.3}$ cm^-2, respectively. The results indicate a flattening in the slope of f(N,X) between the SLLS and DLA. We find little evidence for redshift evolution in the shape of f(N,X) for the SLLS over the redshift range of the sample 1.68 < z < 4.47 and only tentative evidence for evolution in the zeroth moment of f(N,X), the line density l_lls(X). We introduce the observable distribution function O(N,X) and its moment, which elucidates comparisons of HI absorbers from the Lyman-alpha through to the DLA. We find that a simple three parameter function can fit O(N,X) over the range 17.0 <= log(NHI) <=22.0. We use these results to predict that f(N,X) must show two additional inflections below the SLLS regime to match the observed f(N,X) distribution of the Lyman-alpha forest. Finally, we demonstrate that SLLS contribute a minor fraction (~15%) of the universes hydrogen atoms and, therefore, an even small fraction of the mass in predominantly neutral gas.
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