Let $\gamma_{k,s}(n)$ be the number of representations of $n$ as a sum of $s$ distinct $k$-th powers. Clearly $\gamma_k(n)\geq \gamma_{k,s}(n)$ for all $s$. By standard results on Waring's problem, $\gamma_{k,s}(n)\gg n^{s/k-1}$ holds when $s$ is sufficiently large in terms of $k$ (e.g. $s>2^k$), for $n$ sufficiently large in terms of $s$ and $k$. It follows that $\max_{1\leq i\leq n}\gamma_k(i)$ grows faster than any polynomial of $n$ as even the individual terms (with the exception of a bounded number of terms) have this property.

Let $\gamma_{k,s}(n)$ be the number of representations of $n$ as a sum of $s$ distinct $k$-th powers. Clearly $\gamma_k(n)\geq \gamma_{k,s}(n)$ for all $s$. By standard results on Waring's problem, $\gamma_{k,s}(n)\gg n^{s/k-1}$ holds when $s$ is sufficiently large in terms of $k$ (e.g. $s>2^k$), for $n$ sufficiently large in terms of $s$ and $k$. It follows that $\max_{1\leq i\leq n}\gamma_k(i)$ grows faster than any polynomial of $n$ as even the individual terms (with the exception of a bounded number of terms depending on the polynomial) have this property.