Is the following theory countably axiomatizable? Let $L$ be the language of set theory, and for each countable ordinal $\kappa$, define $L_\kappa$ as follows: $L_0 = L$, and $L_{\alpha+1}$ is obtained by adding countably many new class constants $c_\phi$ to the signature of $L_\alpha$ to stand for each formula $\phi(x)$ of one free variable. Define $L_\alpha$ for $\alpha$ a limit ordinal to be the union of $L_\beta$ for $\beta < \alpha$. Given some set theory such as ZF, it is possible to add axioms for the new constants: let $T_0 = ZF$ and $T_{\alpha+1}$ be the theory got from $T_\alpha$ by adding $$x \in c_\phi \iff \phi(x)$$ as an axiom where $\phi(x)$ ranges over the countably many formulas in $L_\alpha$. Let $T_\alpha$ for $\alpha$ a limit ordinal be the union of $T_\beta$ for $\beta<\alpha$. This is ordinary expansion by definitions.
So, for each countable ordinal $\kappa$, $T_\kappa$ is countably axiomatizable. Define $L_\Omega$ and $T_\Omega$ as the union of $L_\kappa$ and $T_\kappa$ as $\kappa$ ranges over countable ordinals.
Is $T_\Omega$ countably axiomatizable?
What relation does MK have to $T_\Omega$, if any?
New question: is every formula of $L_\Omega$ equivalent to a formula of $L_1$ modulo $T_1$? (seems likely) please see: Is every formula of LΩ equivalent to a formula of L1 modulo T1?
 A: As Emil points out, $T_\Omega$ is not countably axiomatizable for trivial reasons: it has an uncountable language, and says non-trivial things about each symbol in that language.
That said, it is equivalent to a countably axiomatizable theory, in a precise sense: namely, you don't really get anything new after $T_\omega$. To see why, think about $T_{\omega+1}$: every formula in $L_\omega$ is also in $L_n$ for some finite $n$, so you've already added a constant symbol naming the relevant class; this means each constant symbol you add at stage $\omega+1$ is - provably in $T_{\omega+1}$ - equivalent to an already-added constant symbol. (Incidentally, this makes it easy to show that MK is much stronger than $T_\omega$, and hence much stronger "morally" than $T_\Omega$.)

In fact, unless I'm missing something your hierarchy collapses right away - $T_1$ is essentially the same as $T_0$. This is because, in forming $T_1$, we can replace every "new" sentence with constant symbols $c_\varphi$ by an "old" sentence where "$-\in c_\varphi$" is replaced with "$\varphi(-)$".
