The Knuth arrow notation is most often defined only on the natural numbers, but it seems that the central idea of it
can be naturally extended to the ordinals, as follows:

$$\alpha\uparrow^0\beta=1$$
$$\alpha\uparrow^1\beta=\alpha^\beta$$
$$\alpha\uparrow^\eta\beta=\sup_{\eta'\lt\eta}[\alpha\uparrow^{\eta'}\sup_{\beta'\lt\beta}(\alpha\uparrow^\eta\beta')].$$

This definition simply replaces the use of $n-1$ and $b-1$ in the usual natural number definition of the Knuth arrow with a supremum over all smaller values, which allows the definition to work with limit ordinals, which have no immediate predecessor. On finite and successor values, this definition agrees with the standard formula. (I'm not sure if others have proposed a different transfinite extension of the concept....I can imagine alternative definitions that treat the limit cases differently, but this won't affect the main conclusion.)

Using this definition, one can show by transfinite induction
that if $\alpha, \beta$ and $\eta$ are countable ordinals, then
$\alpha\uparrow^\eta\beta$ is also countable, because by the
induction hypothesis, this will be a countable supremum of
countable ordinals.

In particular, $\omega\uparrow^\omega\omega$ is a very large
countable ordinal, and the anwer to your question is that it has
cardinality $\aleph_0$.