  We define several notions of a limit point on sequences with domain a barrier
in $[\omega]^{<\omega}$ focusing on the two dimensional case $[\omega]^2$. By
exploring some natural candidates, we show that countable compactness has a
number of generalizations in terms of limits of high dimensional sequences and
define a particular notion of $\alpha$-countable compactness for
$\alpha\leq\omega_1$. We then focus on dimension 2 and compare 2-countable
compactness with notions previously studied in the literature. We present a
number of counterexamples showing that these classes are different. In
particular assuming the existence of a Ramsey ultrafilter, a subspace of
$\beta\omega$ which is doubly countably compact whose square is not countably
compact, answering a question of T. Banakh, S. Dimitrova and O. Gutik. The
analysis of this construction leads to some possibly new types of ultrafilters
related to discrete, P-points and Ramsey ultrafilters.
