Recognizing that $A\cup B$ contains elements in $A$ or $B$, $\mathscr{P}(A\cup B)$ contains subsets of the union, and $\mathscr{P}(\mathscr{P}(A\cup B))$ contains the ordered pair $(a,b)$ for some $a\in A, b\in B$, $\mathscr{P}(\mathscr{P}(\mathscr{P}(A\cup B)))$ contains the Cartesian product $A\times B$ and also the function $f:A\rightarrow B$, $\mathscr{P}(\mathscr{P}(\mathscr{P}(\mathscr{P}(A\cup B))))$ contains the set of all functions from $A$ to $B$ denoted as $B^A$. Until now, everything makes sense mathematically. I wonder if we again construct the power set of the last set, which is $\mathscr{P}^5(A\cup B)$, is there some useful meaning that we can attach to it (along the line of primitive set-theoretic relationship of belonging)?
Sorry if the question is not well-stated...I will try to explain as best as I can.
Further clarification: of course, we can arbitrarily raise the power of the operation $\mathscr{P}$ if we consider ordered triples, quadruples, and so on from multiple sets and build more complex relationships on them. But I wonder if we can take it further with just two sets, as stated in the question - A and B.