Torsion Subgroup (Just a set) for an abelian (non abelian) group.

An example: $G=\langle a,b\mid a^2=b^2=1\rangle$. In it $|ab|=\infty$.

An example that immediately comes to mind is the infinite dihedral group. As motions of the plane this is generated by $s_1$ = reflection w.r.t. the line $x=0$, and by $s_2$ = reflection w.r.t. the line $x=1$. The composition $s_1\circ s_2$ is translation (in the direction of negative $x$-axis) by two units, and thus has infinite order in spite of the two reflections obviously being of order 2.

You may see the answer through the group $\textbf{GL}(2,\mathbb Q)$ and the following members:

$$A=\begin{pmatrix} 0 & -1 \\ 1 & 0 \end{pmatrix},~~B=\begin{pmatrix} 0 & 1 \\ -1 & -1 \end{pmatrix}$$

It can easily be found out that $A^4=E=B^3$ but $|AB|=\infty$.

Source: Rotman's Theory of Group page 27.