When thinking about subalgebras of $\ell_\infty$, the algebra of bounded, scalar-valued, I came across the following question related to counting subalgebras with weak*-separable dual ball obtainable by restricting to closed subsets of the spectrum of $\ell_\infty$.
Let $\beta \mathbb N$ be the Čech–Stone compactification of the discrete space of natural numbers. How many pairwise non-homeomorphic closed and separable subspaces does $\beta \mathbb N$ have?
Such subspaces cannot be second-countable unless finite, but there are many separable closed subspaces (take the closure of any countable subset) and there is no obvious way to classify them up to homeomorphism.
There are exactly $2^c$ pairwise nonhomeomorphic separable closed subspaces in $\beta \mathbb N$.
The proof is essentially just a reference to a great paper:
Since $|\beta \mathbb N| = 2^c$, there are $2^c$ countable subsets in $\beta \mathbb N$, hence $\beta \mathbb N$ has only $2^c$ separable subspaces.
The other, non-trivial, direction is mainly the Main Theorem in A. Dow, A.V. Gubbi, A. Szymanski, "Rigid Stone Spaces within ZFC":
There exist $2^c$ pairwise nonhomeomorphic rigid separable Stone spaces,
where Stone space is (in this paper) an extremally disconnected compact Hausdorff space. As it is well-known, each separable extremally disconnected space can be embedded into $\beta \mathbb N$ (see, for instance, Corollary 3.2 here).
Remark: Perhaps there might be much easier constructions, since here we don't need that the spaces are rigid (= the only autohomeomorphism is the identity).