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.