What is an example of an amenable group $G$ such that $G^\mathbb{N}$ isn't amenable?
My thoughts: Consider for example $G=\{\begin{pmatrix}a&b\\0&1\end{pmatrix}\ |\ a,b\in\mathbb{R}, a\neq 0\}$, which is amenable as it is solvable. Now if we could prove that for every $n$ there are $g_n,h_n\in G$ such that every non trivial word in $g_n,h_n$ (viewed as if in the free group) which is equal in $G$ to the identity must have length at least $n$, then $(g_n)_{n\in\mathbb N}$ and $(h_n)_{n\in\mathbb N}$ would generate a free group in $G^\mathbb{N}$. Is it possible to find such $g_n,h_n$? If not, how could we proceed?
Hint: First, find an amenable group $G$ which contains (isomorphic copies of) all finite groups.