  We define a hierarchy of systems with topological completely positive entropy
in the context of continuous countable amenable group actions on compact metric
spaces. For each countable ordinal we construct a dynamical system on the
corresponding level of the aforementioned hierarchy and provide subshifts of
finite type for the first three levels. We give necessary and sufficient
conditions for entropy pairs by means of the asymptotic relation on systems
with the pseudo-orbit tracing property, and thus create a bridge between a
result by Pavlov and a result by Meyerovitch. As a corollary, we answer
negatively an open question by Pavlov regarding necessary conditions for
completely positive entropy.
