  Lyman Limit systems (LLSs) trace the low-density circumgalactic medium and
the most dense regions of the intergalactic medium, so their number density and
evolution at high redshift, just after reionisation, are important to
constrain. We present a survey for LLSs at high redshifts, $z_{\rm LLS}
=3.5$--5.4, in the homogeneous dataset of 153 optical quasar spectra at $z \sim
5$ from the Giant Gemini GMOS survey. Our analysis includes detailed
investigation of survey biases using mock spectra which provide important
corrections to the raw measurements. We estimate the incidence of LLSs per unit
redshift at $z \approx 4.4$ to be $\ell(z) = 2.6 \pm 0.4$. Combining our
results with previous surveys at $z_{\rm LLS} <4$, the best-fit power-law
evolution is $\ell(z) = \ell_* [(1+z)/4]^\alpha$ with $\ell_* = 1.46 \pm 0.11$
and $\alpha = 1.70 \pm 0.22$ (68\% confidence intervals). Despite hints in
previous $z_{\rm LLS} <4$ results, there is no indication for a deviation from
this single power-law soon after reionization. Finally, we integrate our new
results with previous surveys of the intergalactic and circumgalactic media to
constrain the hydrogen column density distribution function, $f(N_{\rm HI},X)$,
over 10 orders of magnitude. The data at $z \sim 5$ are not well described by
the $f(N_{\rm HI},X)$ model previously reported for $z \sim 2$--3 (after
re-scaling) and a 7-pivot model fitting the full $z \sim 2$--5 dataset is
statistically unacceptable. We conclude that there is significant evolution in
the shape of $f(N_{\rm HI},X)$ over this $\sim$2 billion year period.
