(Now asked at MO.)
Given a topological space $\mathcal{X}$, let $$\mathsf{Cl_C}(\mathcal{X})=\bigcup_{n\in\mathbb{N}}C(\mathcal{X}^n,\mathcal{X})$$ be the clone of finite-arity continuous functions on $\mathcal{X}$ (where $\mathcal{X}^n$ is given the product topology). Motivated by the Kolmogorov-Arnold superposition theorem, let the superposition dimension of $\mathcal{X}$ be the minimal cardinality of a set of continuous functions $\mathfrak{F}\subseteq\mathsf{Cl_C}(\mathcal{X})$ such that the clone generated by $\mathfrak{F}\cup C(\mathcal{X},\mathcal{X})$ is all of $\mathsf{Cl_C}(\mathcal{X})$.
Some quick observations:
Any space homeomorphic to its square trivially has superposition dimension $1$: there is no real distinction between single- and multi-variable continuous functions over such a space once we bring in an appropriate pairing function.
The K/A theorem says that the superposition dimension of $[0,1]$ is $1$, witnessed by $\{+\}$.
- Admittedly the failure to distinguish between Cantor space and the unit interval suggests that the word "dimension" is being wildly misused, but meh.
It is not clear to me that shifting attention to maps with codomain $[0,1]$ results in the same notion (I strongly suspect it doesn't), so e.g. Ostrand's extension of the K/A theorem doesn't seem relevant here.
I'm generally curious for any relevant information, but the following particular question seems interesting: is there, for each finite $n$, a "nice" topological space with superposition dimension $n$? Of course this is really a whole family of questions, one for each meaning of "nice." Tentatively I think the most interesting case is likely to be the following:
Question: Is there, for each finite $n$, a connected topological manifold (with or without boundary, not necessarily compact) with superposition dimension $n$?
I suspect that every topological manifold has superposition dimension $1$, by some not-too-complicated extension of K/A, but I don't immediately see how to show that.