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?
Best Answer
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.