Here is a collection of what I have so far thanks to the answers by Guntram and S. Carnahan. Let's denote by $P(n,m)$ the property that $|A^m| \le\binom{n+m-1}{m}$ for all subsets $|A|=n$.

We have that the only nonabelian $P(2,2)$ groups are of the form $Q_8\times G$ where $G$ is an elementary abelian 2-group, and that $P(3,2)$ groups have to be abelian by Freiman's paper "On two- and three-element subsets of groups".

In "A characterization of abelian groups", Brailovsky proves that large enough $P(n,2)$ are abelian by showing that $P(n,2)\implies P(n',2)$ for all $n\geq n'\geq 2$, so that the result follows from the previous paragraph.

In "Small squaring and cubing properties for finite groups", Berkovich, Freiman and Praeger prove that the only nonabelian group with $P(2,3)$ is $S_3$.

On the other hand there are nonabelian groups with $P(n,m)$ whenever $\binom{n+m-1}{m}\geq 2^{2n+1}$ as in Guntram's answer.

Here is a collection of what I have so far thanks to the answers by Guntram and S. Carnahan. Let's denote by $P(n,m)$ the property that $|A^m| \le\binom{n+m-1}{m}$ for all subsets $|A|=n$.

We have that the only nonabelian $P(2,2)$ groups are of the form $Q_8\times G$ where $G$ is an elementary abelian 2-group, and that $P(3,2)$ groups have to be abelian by Freiman's paper "On two- and three-element subsets of groups".

In "A characterization of abelian groups", Brailovsky proves that large enough $P(n,2)$ are abelian by showing that $P(n,2)\implies P(n',2)$ for all $n\geq n'\geq 2$, so that the result follows from the previous paragraph.

In "Small squaring and cubing properties for finite groups", Berkovich, Freiman and Praeger prove that the only nonabelian group with $P(2,3)$ is $S_3$.

On the other hand there are nonabelian groups with $P(n,m)$ whenever $\binom{n+m-1}{m}\geq 2^{2n+1}$ as in Guntram's answer.