In real analysis, subsets of $\mathcal{P}^n(\mathbb{R})$ arise a lot, for many different choices of $n \in \mathbb{N}$. However, I've never seen a 'multi-level' set, like I've never seen a subset of $\mathbb{R} \cup \mathcal{P}(\mathbb{R})$ that can be expressed in the form $A \cup B$, where $A$ and $B$ are non-empty sets satisfying $A \subseteq \mathbb{R}$ and $B \subseteq \mathcal{P}(\mathbb{R}).$
Along a similar vein, I have never seen anything like $\bigcup_{n \in \mathbb{N}} \mathcal{P}^n(\mathbb{R}).$ Are there situations where these kinds of 'multi-level' collections arise naturally, outside of set-theory? I'm not insisting the base case be $\mathbb{R}$ - that was just an example.
Anyway, here's a related question. Is it critical for the rest of mathematics (outside of set theory) that we allow the construction of such collections?
In geometry, one often thinks of a line (or circle or ...) as a set of points, but one also wants to consider a configuration as a set of points and lines. So a configuration is a two-level set.
If one represents functions as sets of ordered pairs and represents ordered pairs as sets in the usual way, then any function from elements of a set $X$ to subsets of $X$ (or vice versa) will involve (inside the ordered pairs) two-level sets. More levels arise in the case of functions that move things up or down by more than one step in the set-theoretic hierarchy, for example the function that sends a point in $\mathbb R$ to the Dirac measure concentrated at that point.