18.83 (2014)

A generating system $X$ of a group $G$ is fast if there is an integer $n$ such that every element of $G$ can be expressed as a product of at most $n$ elements of $X$ or their inverses. If not, we say that it is slow. For instance, in $(\mathbb{Z}, +)$, the squares are fast, but the powers of 2 are slow.
Do there exist countable infinite groups without an infinite slow generating set? Uncountable ones do exist.