  Given two elementary embeddings from the collection of sets of rank less than
$\lambda$ to itself, one can combine them to obtain another such embedding in
two ways: by composition, and by applying one to (initial segments of) the
other. Hence, a single such nontrivial embedding $j$ generates an algebra of
embeddings via these two operations, which satisfies certain laws (for example,
application distributes over both composition and application). Laver has
shown, among other things, that this algebra is free on one generator with
respect to these laws.
  The set of critical points of members of this algebra is the subject of this
paper. This set contains the critical point $\kappa_0$ of $j$, as well as all
of the other ordinals $\kappa_n$ in the critical sequence of $j$ (defined by
$\kappa_{n+1} = j(\kappa_n)$). But the set includes many other ordinals as
well. The main result of this paper is that the number of critical points below
$\kappa_n$ (which has been shown to be finite by Laver and Steel) grows so
quickly with $n$ that it dominates any primitive recursive function. In fact,
it grows faster than the Ackermann function, and even faster than a slow
iterate of the Ackermann function. Further results show that, even just below
$\kappa_4$, one can find so many critical points that the number is only
expressible using fast-growing hierarchies of iterated functions (six levels of
iteration beyond exponentials).
