Q:

Example of a group where $o(a)$ and $o(b)$ are finite but $o(ab)$ is infinite

Let G be a group and $a,b \in G$. It is given that $o(a)$ and $o(b)$ are finite.  Can you give an example of a group where $o(ab)$ is infinite? 

A:

Probably you have seen this effect: mirrors on two opposite walls of a room.   
 
Take two parallel hyperplanes in $\mathbb R^n$.  Reflection in each of them is an isometry of order 2.   But their composition is a translation of infinite order. 

A:

Consider the matrices $$A=\left(\begin{array}[cc] .1 & -1\\ 0 & -1\end{array}\right)\quad\text{and}\quad B=\left(\begin{array}[cc].1&0\\ 0 & -1\end{array}\right).$$ You can check that each have order $2$, but their product gives $$(AB)^n=\left(\begin{array}[cc] .1 & n\\ 0 & 1\end{array}\right),$$a matrix without finite order. 

A:

The standard example is the infinite dihedral group. 
Consider the group of maps on $\mathbf{Z}$ 
$$
D_{\infty} = \{ x \mapsto \pm x + b : b \in \mathbf{Z}  \}. 
$$
Consider the maps
$$
\sigma: x \mapsto -x,
\qquad
\tau: x \mapsto -x + 1,
$$
both of order $2$. Their composition
$$
\tau \circ \sigma (x) = \tau(\sigma(x)) : x \mapsto x + 1
$$
has infinite order.