How does one count cardinalities of sets of sets of sets of ...?
What's the cardinality of
$$\{ \{ \{ x\}\}\}$$
Also if it's only counted from the outermost or any particular "level", then what kind of sense does this kind of cardinality have? Why wouldn't the cardinality contain the "depth count" as well?
Also, since there's the rank for counting the depth of the set, are there connections between ranks and cardinalities?
Physically it seems that it makes sense to think of cardinality of set as the number of distinct objects, but depending on what one considers as objects, then it could contain the lower most stuff. E.g. if one thinks of something as a set of particles making that thing up, then while they may also have inclusions, then they also make up the whole object, so the separation between rank and cardinality becomes more blurry.
There is exactly one way to define cardinality: it's the size of a set. What are the elements of the set is irrelevant. The sets $\Bbb N$ and $\Bbb Q$ have the same cardinality, $\aleph_0$, regardless to what are the exact objects that sit inside each set.
In particular, the set $\{\{\{x\}\}\}$ has one element, $\{\{x\}\}$, and therefore its cardinality is exactly $1$.
What you are thinking about is the transitive closure, perhaps. Which in this case would be $\{x,\{x\},\{\{x\}\}\}$, which has three members and therefore has cardinality $3$. At least assuming that $x=\varnothing$, otherwise we need to throw the transitive closure of $x$ into the mix.