A group $G$ is said to be boundedly generated if (it is finitely generated and) there exists a finite family of cyclic subgroups (not necessarily normal or distinct) $\lbrace C_i \rbrace_{i =1, \ldots, k}$ such that $G = C_1 \cdots C_k$.

Simplest examples of boundedly generated groups are finite groups and polycyclic groups. Free groups are not boundedly generated, and $SL_2(\mathbb{Z})$ also isn't (since passing to finite index subgroups does not change this property).

*A priori*, the order in the product matters. Assume $G = C_1 \cdots C_k$ is a boundedly generated group so that $G = C_{\sigma(1)} \cdots C_{\sigma(k)}$ for any permutation $\sigma \in Sym_k$. In particular, one may then assume the $C_i$ are distinct. Except virtually polycyclic groups, which group has this property?