There has been a lot of discussion here about meta-meta-mathematics really not being a concrete notion, and the meta-meta-level collapsing to the meta-level because a meta-theory is of course, a theory.

However, I think the mistake many have implicitly made in the discussion is to assume that a meta-theory is "just" a theory. If $T_0$ is a theory (Which I will identify with its consequence relation $\vdash_{T_0}$ with a language $L_0$ of formulas, then a *meta-theory* $T_1$ is simply another consequence relation $\vdash_{T_1}$ whose language of formulas $L_1$ includes sentences of the form $"\Gamma \vdash_{T_0} \phi"$ for contexts $\Gamma$ and formulas $\phi$ in the base language $L_0$, such that: $$ \vdash_{T_1} "\Gamma \vdash_{T_0} \phi" \iff \Gamma \vdash_{T_0} \phi $$

So under this definition (which I think matches up with the intuitive idea of what a "metatheory" is, at least in my conception of it), a metatheory is a theory, but not *just* a theory -- it is a theory with additional structure.

Thus, in this sense we can meaningfully talk about meta-meta-theories by treating a meta-theory $T_1$ as a theory, and finding a metatheory $T_2$ for $T_1$ (treated as a theory). This alone would simply be a meta-theory again, but if we keep the structure of the metatheory $T_1$ (instead of throwing it away and simply treating it like a theory), we obtain an honest-to-goodness metametatheory, which would have something like the structure:

$$ ((\vdash_{T_2},L_2),(\vdash_{T_1},L_1),(\vdash_{T_0},L_0)) $$

together with two "quoting operations" as I have described above, letting the metametatheory talk about the consequence relation of the metatheory, and the metatheory talk about the consequence relation of the base theory.

Godel's incompleteness theorem, though as mentioned in François G. Dorais' answer was originally formulated to talk about the "metatheory" PM, did not meaningfully make use of the metatheoretical structure in PM, and thus is *not* an honest-to-goodness metametatheorem, which would make meaningful use of the structure I have described above.

Whether or not such mathematics has been done, I'm not sure. Possibly Carl Mummert's answer gives an example of such of a study which cannot meaningfully be reduced to simply metamathematics, but I'm afraid without knowing the details it is difficult for me to comment definitively.