This question occurred to me while I was writing my most recent answer.
Recall that given an extended metric space $(X,d)$ the hyperspace of $X$, which I'll write as $\def\Hc{\mathcal{H}}\Hc(X)$, is the set of non-empty closed subsets of $X$ endowed with the Hausdorff metric. (Often in the literature there is an additional restriction that $\Hc(X)$ only consists of the compact or at least bounded sets, which in particular ensures that $\Hc(X)$ is a metric space (rather than just an extended metric space) when $X$ is a metric space, but I'd be surprised if that changed the answer of this question.) I'm going to be writing $\Hc^n(X)$ for the $n$-th iterated hyperspace of $X$ (e.g., $\Hc^3(X) = \Hc(\Hc(\Hc(X)))$).
As I mentioned in my linked answer, the Kuratowski pair map $\langle x, y\rangle \mapsto \{\{x\},\{x,y\}\}$ yields a $1$-Lipschitz embedding of $X^2$ (with the $\max$ metric) into $\Hc(\Hc(X))$ for any metric space $X$. This isn't an isometric embedding, however, as the inverse is only guaranteed to be $3$-Lipschitz (which is sharp in the specific case of $X= \mathbb{R}$ with the pairs $\langle 2,0\rangle$ and $\langle 1, 3\rangle$).
If you tweak the definition of hyperspace to allow $\varnothing$ as an element, then it is possible to get an isometric embedding. Specifically, let $\Hc_\varnothing(X)$ be the set of all closed subsets of $X$ endowed with the Hausdorff metric (with the understanding that $d_H(\varnothing,A) = \infty$ for any non-empty $A \subseteq X$). It's now relatively easy to show that the Wiener's older definition of ordered pair $\langle x, y \rangle := \{\{\{x\},\varnothing\},\{\{y\}\}\}$ gives an isometric embedding of $X^2$ into $\Hc^3_\varnothing(X)$.
It seems like this kind of embedding really needs the guaranteed separation of $d_H(\varnothing,A) = \infty$ to be isometric, which leads me to my questions.
Question 1. Is it true that for every extended metric space $X$, there is an $n$ such that $X^2$ (with the $\max$ metric) isometrically embeds into $\Hc^n(X)$?
It seems pretty likely that if this is possible, it will be because of a uniform construction, so it's natural to wonder how uniform it can be.
Note that in the category of extended metric spaces with $1$-Lipschitz maps, $X \mapsto \Hc(X)$ can be regarded as a covariant functor (specifically by taking $f : X \to Y$ to the map $\Hc(f) : \Hc(X) \to \Hc(Y)$ defined by $\Hc(f)(A) = \overline{f[A]}$, where $\overline{f[A]}$ is the closure of the image of $A$ under $f$). Likewise the Cartesian square construction is functorial (taking $f: A \to B$ to $g : A^2 \to B^2$ defined by $g(\langle x,y \rangle) = \langle f(x),f(y)\rangle$). Finally, it's fairly easy to see that the Kuratowski pair map is a natural transformation from $X \mapsto X^2$ to $X \mapsto \Hc^2(X)$ (and, for that matter, the Wiener pair map is a natural transformation from $X \mapsto X^2$ to $X \mapsto \Hc^3_\varnothing(X)$, which is also functorial).
Question 2. Is there an $n$ and a natural transformation $\eta$ from $X \mapsto X^2$ to $X \mapsto \Hc^n(X)$ such that for every extended metric space $X$, $\eta_X : X^2 \to \Hc^n(X)$ is an isometric embedding?
