  We consider the question of how many edge-disjoint near-maximal cliques may
be found in the dense Erd\H{o}s-R\'enyi random graph $G(n,p)$. Recently Acan
and Kahn showed that the largest such family contains only $O(n^2/(\log{n})^3)$
cliques, with high probability, which disproved a conjecture of Alon and
Spencer. We prove the corresponding lower bound, $\Omega(n^2/(\log{n})^3)$, by
considering a random graph process which sequentially selects and deletes
near-maximal cliques. To analyse this process we use the Differential Equation
Method. We also give a new proof of the upper bound $O(n^2/(\log{n})^3)$ and
discuss the problem of the precise size of the largest such clique packing.
