The Knuth arrow notation is most often defined only on the natural numbers, but the central idea of it
can be easily extended to the ordinals, for example as follows:
$$\alpha\uparrow^0\beta=\alpha\beta$$
$$\alpha\uparrow^\eta 0=1\qquad\text{for }\eta\geq 1$$
$$\alpha\uparrow^\eta\beta=\sup_{\eta'\lt\eta}[\alpha\uparrow^{\eta'}\sup_{\beta'\lt\beta}(\alpha\uparrow^\eta\beta')]\qquad\text{otherwise}.$$
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.
Please note that there are other natural definitions, depending on how one treats the limit stage. which will not be the same as this one. For example, in the definition above, by $\alpha\beta$ I had intended that one should use the natural product, rather than the common product, because this achieves some nicer properties. But others may want to do things differently, and the resulting functions will differ. Nevertheless, what I say below will apply to all the natural formulations of the arrow.
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 $\aleph_0$.
