This is essentially equivalent to <a href="http://mathoverflow.net/questions/43964/folner-sequences-of-amenable-groups-of-exponential-growth"> this </a> question by Simon Thomas. Let $G=\langle X\rangle$ be a finitely generated group, $b_n$ be the number of elements in the ball of radius $n$ in the Cayley graph. 

1) Is it possible that the limit $\lim \frac{b_{n+1}}{b_n}$ does not exist? 

2) Suppose that for every constant $\epsilon>0$ there exists an $n$ such that $\frac{b_{n+1}}{b_n}\le 1+\epsilon$. Does it imply that $\lim \frac{b_{n+1}}{b_n}=1$? 

Note that the condition of 2) implies that the group is amenable and one can take balls as Foelner sets (which would contradict a statement in de la Harpe's book).