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.