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If $cd(S_n)$ is the set of degrees of irreducible, complex-valued characters on the symmetric group $S_n$, then by the basic representation theory of the symmetric group this is the same as the set $\{f^{\lambda}\colon \lambda\vdash n\}$, where for a partition $\lambda\vdash n$, $f^{\lambda}$ is the number of Standard Young Tableaux of shape $\lambda$, which by the hook-length formula has the explicit form $f^{\lambda}=n! \cdot \prod_{u \in \lambda}\frac{1}{h_u!}$. I would imagine that the number of pairs of partitions $\lambda,\nu \vdash n$ with $f^{\lambda}=f^{\nu}$ is very small as a fraction of all pairs of partitions of $n$. And probably that would not be hard to formally establish (for instance using the hook-length formula). So the size $|cd(S_n)|$ of the set of degrees of irreps should be basically the same as the number of partitions on $n$. This is a very famous quantity whose asymptotics are well-understood (see e.g. https://en.wikipedia.org/wiki/Partition_function_(number_theory) as a starting point).