Update: I've reversed the inequality.. sorry for the mistake!
Let $G$ be a finitely generated group with a fixed symmetric generating set $S$. Let $d_S$ be the word distance with respect to $S$ and let $B(e,R)$ and $S(e,R)$ denote the open ball and the sphere around the unit element $e$ with radius $n\in\mathbb N$.
Question: Does there exist a non-amenable group such that $|B(e,n)|>|S(e,n)|$ for all but finitely many $n$'s?
I hope it is not too easy... I'm trying to use $\mathbb F_2\times(\mathbb Z_2)^s$, with $s$ big enough, but I'm quite confused at the moment and I'm not convinced one can do the job with a group containing $\mathbb F_2$. The balls increase too quickly and even if you try to make their growth slower using a hugely generated amenable group, it seems to me that at a certain point $\mathbb F_2$ wins.. mmm sorry for the informal language of this sentence. I hope you have understood what I meant.
Thanks in advance for any comment/suggestion