  This paper deals with countable products of countable Borel equivalence
relations and equivalence relations "just above" those in the Borel
reducibility hierarchy. We show that if $E$ is strongly ergodic with respect to
$\mu$ then $E^\mathbb{N}$ is strongly ergodic with respect to $\mu^\mathbb{N}$.
We answer questions of Clemens and Coskey regarding their recently defined
$\Gamma$-jump operations, in particular showing that the $\mathbb{Z}^2$-jump of
$E_\infty$ is strictly above the $\mathbb{Z}$-jump of $E_\infty$. We study a
notion of equivalence relations which can be classified by infinite sequences
of "definably countable sets". In particular, we define an interesting example
of such equivalence relation which is strictly above $E_\infty^\mathbb{N}$,
strictly below $=^+$, and is incomparable with the $\Gamma$-jumps of countable
equivalence relations.
  We establish a characterization of strong ergodicity between Borel
equivalence relations in terms of symmetric models. The proofs then rely on a
fine analysis of the very weak choice principles "every sequence of $E$-classes
admits a choice sequence", for various countable Borel equivalence relations
$E$.
