  For a continuous map $T$ of a compact metrizable space $X$ with finite
topological entropy, the order of accumulation of entropy of $T$ is a countable
ordinal that arises in the context of entropy structure and symbolic
extensions. We show that every countable ordinal is realized as the order of
accumulation of some dynamical system. Our proof relies on functional analysis
of metrizable Choquet simplices and a realization theorem of Downarowicz and
Serafin. Further, if $M$ is a metrizable Choquet simplex, we bound the ordinals
that appear as the order of accumulation of entropy of a dynamical system whose
simplex of invariant measures is affinely homeomorphic to $M$. These bounds are
given in terms of the Cantor-Bendixson rank of $\overline{\ex(M)}$, the closure
of the extreme points of $M$, and the relative Cantor-Bendixson rank of
$\overline{\ex(M)}$ with respect to $\ex(M)$. We also address the optimality of
these bounds.
