  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 universe's hydrogen atoms and, therefore, an even small
fraction of the mass in predominantly neutral gas.
