Let $G$ be a discrete and finitely generated group. Recall that $\{F_n\}_{n \in \mathbb{N}}$ is a Følner sequence if $|g F_n \cup F_n|/|F_n| \rightarrow 1$ for every $g \in G$. As is well known, existence of a Følner sequence is equivalent to amenability of $G$.

It is often said that Følner sequences have strange shapes. My soft question is: which examples do we have that support this claim? Of course, if $G$ is of subexponential growth then a subsequence of balls forms a Følner sequence, and this does not have a weird shape. Hence, more specifically: which examples of groups of exponential growth do we know that have explicit Følner sequences not made of balls?

As instances of the examples I am asking for, Star-shaped Folner sequence asks for Følner sets of a certain form, while an answer of Folner sets and balls gives explicit sequences made of rectangles (as opposed to balls). Likewise, the ax + b group has a Følner sequence made of rectangles where one side is exponentially larger than the other.