  We show that if $\kappa \leq \omega$ and there exists a group topology
without non-trivial convergent sequences on an Abelian group $H$ such that
$H^n$ is countably compact for each $n<\kappa$ then there exists a topological
group $G$ such that $G^n$ is countably compact for each $n <\kappa$ and
$G^{\kappa}$ is not countably compact. If in addition $H$ is torsion, then the
result above holds for $\kappa=\omega_1$. Combining with other results in the
literature, we show that:
  $a)$ Assuming ${\mathfrak c}$ incomparable selective ultrafilters, for each
$n \in \omega$, there exists a group topology on the free Abelian group $G$
such that $G^n$ is countably compact and $G^{n+1}$ is not countably compact.
(It was already know for $\omega$).
  $b)$ If $\kappa \in \omega \cup \{\omega\} \cup \{\omega_1\}$, there exists
in ZFC a topological group $G$ such that $G^\gamma$ is countably compact for
each cardinal $\gamma <\kappa$ and $G^\kappa$ is not countably compact.
