Examples of infinite groups such that all their respective elements are of finite order. I am in need of examples of infinite groups such that all their respective elements are of finite order.

Let $G=\bigoplus_{n=1}^\infty(\mathbb{Z}/n\mathbb{Z})$.
By definition, each $g=(g_i)_{i=1}^\infty\in G$ satisfies $g_i=0$ for all but finitely many $i$, thus $g$ has finite order.
However, it has infinite exponent: Consider $(0,\dots,1,\dots,0)$ with 1 in the $j$th position; it has order $j$.
