Interesting meta-meta-mathematical theorems? The Goedel incompleteness theorems can be considered meta-mathematical theorems, as they are "written" in a meta-theory and "talk" about properties of a class of formal theories.
The following may be a naive question, but...

Are there any "interesting" results at the next level, i.e. so to
  speak, that take place in a meta-meta-theory and talk about
  meta-theories and properties thereof and the theories they describe/codify?

 A: In Reverse Mathematics, we can study what happens if we use weak systems of second-arithmetic as metatheories. For example, we can study the strength of the completeness theorem and prove results such as "Gödel’s completeness theorem is equivalent to $\mathsf{WKL}_0$ over $\mathsf{RCA}_0$." That can be seen as a meta-meta-theorem: we are investigating which axioms are required in the metatheory for the completeness theorem to hold. 
This is not as trivial as it may sound; some results are genuinely unexpected. For example, one interesting fact is that every countable $\omega$-model $M$ of $\mathsf{WKL}_0$ contains a real $C$ that codes a countable $\omega$-model of $\mathsf{WKL}_0$. Due to other weaknesses of $\mathsf{WKL}_0$, this does not cause $\mathsf{WKL}_0$ to be inconsistent! We identify the coded $\omega$-model $C$ not within $M$, but at a level one step above $M$; the model $M$ will not, in general, recognize that $C$ satisfies $\mathsf{WKL}_0$. So we are viewing $\mathsf{WKL}_0$ as our metatheory and our object theory, but not as our meta-meta-theory - we cannot prove the desired result in $\mathsf{WKL}_0$ because of incompleteness phenomena.
A: This answer is intended to clarify my comments to Sébastien's answer and also to propose a properly meta-meta-fact.
There is an intrinsic problem with the idea of meta-meta-theorems because theorems are mathematical ideas and therefore talking about them belongs in the meta-theory and hence cannot be properly meta-meta-theoretical. It's true that statements like "the meta-theory is incomplete" are, as stated, meta-meta-theoretical but once you sit down to formalize (I wish I could say "mathematicize") what that statement means, it suddenly loses its meta-meta-theoretical flavor and it becomes simply meta-mathematical or even plainly mathematical.
That doesn't mean that meta-meta-facts don't exist, they just need to involve ideas that are impossible to formalize in a mathematical sense. One such idea is the following variant of the sorites paradox. The understandable numbers have the following two properties:


*

*$0$, $1$, $2$ are understandable and if $n$ is understandable then so is $n+1$.

*The Ackermann number $A(5,5)$ is not understandable.


There is no mathematical concept that corresponds to understandable numbers since mathematical concepts obey mathematical induction and that contradicts the two properties above. However, the concept of understandable numbers still makes sense. That fact — understandable numbers make sense — is properly meta-meta-mathematical, though I would hardly call this a meta-meta-theorem since I can't imagine how I could prove this.
You don't have to go that far beyond mathematics to come across a meta-meta-mathematical statement. The statement "the meta-theory is incomplete" that I mentioned earlier is meta-meta-theoretical in intent. The similar statement "any meta-theory is incomplete" is even more clearly meta-meta-theoretical. This last statement is true in a practical sense since any practical meta-theory should interpret arithmetic and should be computably axiomatizable, but such theories are incomplete by Gödel's theorem. However, it is not really true that "any meta-theory is incomplete" since, for example, the theory of true arithmetic is complete and perfectly usable as a meta-theory, but the drawback is that we don't understand what the axioms of this theory actually are. As I just illustrated, depending on how you choose to formalize "any meta-theory is incomplete", you may get different answers. Each such answer is a meta-theorem but not a meta-meta-theorem because of the formalization process which involves a mathematical interpretation of the statements. To avoid this meta-collapse, you must resist the temptation to give the meta-meta-statement any concrete mathematical sense. The catch is that you can't really prove such meta-meta-statements without first transforming them into mathematical statements, so there is little hope in finding a meta-meta-theorem in any proper sense.

To further illustrate the issue, note that Gödel's Incompleteness Theorems were originally meta-meta-theorems: Gödel proved that the formal system of Principia Mathematica (PM) was incomplete and PM was intended by Russell and Whitehead as the foundation of all mathematics, i.e., the ultimate meta-theory. Today, we understand Gödel's results as meta-theorems that apply to any computably axiomatizable theory that can interpret enough arithmetic, regardless of whether such theories are thought of as meta-theories.
This collapse of meta-levels is systematic. A meta-meta-theorem is just a meta-theorem applied in the context of a meta-theory, and any meta-theorem applied in the context of a meta-theory is a meta-meta-theorem. Since the difference between a theory and a meta-theory is only one of intent, there is no concrete distinction between meta-theorems and meta-meta-theorems.
A: You could let $\alpha_0$ be the statement Con(ZFC), and $\alpha_{n+1}$ be ZFC $\not\vdash\alpha_n$, and at limit ordinals $\alpha_\lambda$ is $(\forall \beta<\lambda)($ZFC $\not\vdash \alpha_\beta)$. Then for each computable ordinal $\lambda$, $\alpha_\lambda$ is true assuming $\alpha_0$ is. Which is a theorem of meta$^{\omega_1^{\text{CK}}}$-mathematics. :)
But we can view theoretical computer science, theoretical physics, mathematical finance, and in general the discipline studying mathematical descriptions of any phenomenon, as subsets of mathematics.
In which case metamathematics, being the mathematical description of mathematics, is also a subset of mathematics (and therefore meta$^\lambda$-mathematics is a subset of meta-mathematics).
(Acknowledgments: This answer is indebted to answer and comments of Sébastien and François.)
A: There are statements which are  independent but not provably independent
If the independence of a statement is a meta-mathematical theorem, then the existence of statements which are independent but not provably independent is a meta-meta-mathematical theorem.
See the post:  Are there statements that are undecidable but not provably undecidable      (undecidable is here synonymous of independent) and the positive answer (under ZFC, assuming its consistence).
A: If the question can be phrased like this “Is it possible to obtain a result  in meta theory of a theory, which cannot be obtained in that theory”, then I believe the answer is “No”, and this can be proved in strict terms of mathematical logic. I did not look into detail, but I believe it is easy to prove the following 
Theorem. For any theory T in language L and a metatheory T’ of T in language L’,  such that L and L’ do not have symbols in common,  there exists a common conservative extension of T and T'. 
This theorem shows that whenever we obtain an interesting result in a meta theory T’ of a theory T, we can obtain same result in a conservative extension of T.  In order to apply the theorem and get to this,  we first rename all symbols in the axioms of T (and in the language of T’) so that the languages of T and T' do not have symbols in common, then apply the theorem.
A: 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.
