To "specify" Alain's answer: 0) The group $\langle a,b \mid bab^{-1}=a^2\rangle$ (solvable of class 2 Baumslag-Solitar group) 1) The group of upper triangular $n\times n$ matrices with integer coefficients and 1 on the diagonal (nilpotent), $n\ge 1$. 2) The group of all permutations of $\mathbb{Z}$ with finite support (locally finite). 3) The subexp. growth groups, unfortunately, would require more space to define. But you can find them in Wiki.

To "specify" Alain's answer: 1) The group of upper triangular $n\times n$ matrices with integer coefficients and 1 on the diagonal (nilpotent), $n\ge 1$. 2) The group of all permutations of $\mathbb{Z}$ with finite support (locally finite). 3) The subexp. growth groups, unfortunately, would require more space to define. But you can find them in Wiki.