Let $G$ be a non-elementary group generated by a finite set $S$. Here, a group is called non-elementary if it is not virtually abelian. Denote $S^{\le n}:=\{g\in G: |g|_S\le n\}$ for any $n\in \mathbb N$ and $[H,K]:=\{[h,k]: h\in H, k\in K\}$ for any two subsets $H,K\subset G$. It is easy to see that $|S^{\le n}|\ge n$ as $G$ is infinite. After some research, I know that the whole commutator subset $[G,G]$ is still infinite. So my question is that whether there is a strictly increasing function $\kappa: \mathbb N_{\ge 1}\to \mathbb N_{\ge 1}$ such that $|[S^{\le n}, S^{\le n}]|\ge \kappa(n)$ for any $n\in \mathbb N_{\ge 1}$?

Maybe a weaker question is that: suppose more that $G$ is not nilpotent, then $\kappa(n)$ can be chosen like $c\log(n)$?

Let $G$ be a non-elementary group generated by a finite set $S$. Here, a group is called non-elementary if it is not virtually abelian. Denote $S^{\le n}:=\{g\in G: |g|_S\le n\}$ for any $n\in \mathbb N$ and $C(H,K):=\{[h,k]: h\in H, k\in K\}$ for any two subsets $H,K\subset G$. It is easy to see that $|S^{\le n}|\ge n$ as $G$ is infinite. After some research, I know that the whole commutator subset $C(G,G)$ is still infinite. So my question is that whether there is a strictly increasing function $\kappa: \mathbb N_{\ge 1}\to \mathbb N_{\ge 1}$ such that $|C(S^{\le n}, S^{\le n})|\ge \kappa(n)$ for any $n\in \mathbb N_{\ge 1}$?

Maybe a weaker question is that: suppose more that $G$ is not nilpotent, then $\kappa(n)$ can be chosen like $c\log(n)$?