Examples of a torsion group where there are elements of arbitrary large order.

Question

So, I wish to find examples of torsion group where there exist elements of arbitrarily large order. That is a group $G$ where for each $n\in\mathbb{N}$, there exists $g\in G$ such that $g^n = e$ (operation of $G$ applied to $g$ $n$-times), where $e$ is the identity element. I could find one example, whose existence might seem to involve Axiom of Choice. Consider additive groups modulo $n$ and their product over all natural numbers. $$\prod_{n\in\mathbb{N}}\mathbb{Z}_n$$ These are just sequences where $n$-th element is from the additive group $\mathbb{Z}_n$. This is not yet the example, for example $(1,1,1,\ldots,...1,\ldots)$ has infinite order. Consider subgroup $G\leq \prod\mathbb{Z}_n$ of all sequences with finite support. This indeed is a group with elements of arbitrarily large order.

BUT... In order to show there are even elements of $\prod\mathbb{Z}_n$, we must indeed invoke some form of choice. Of course not for elements of $G$. My question is, can some other, possibly simpler, examples be found?

Answer 1

Probably the most neat examples of such groups are quasicyclic groups. Not only they do satisfy your condition, but also they are locally cyclic $p$-groups.

Answer 2

No, no form of choice is needed, as $(0,...,0,...)$ is an element, and $(1,...,1,...)$ as well, and many other examples.

Other examples include, as given in the comments $\mathbb{Q/Z}$, or its "$p$-analogue" $\mathbb Z/p^\infty$, or of course any torsion group containing them (e.g. any direct sum of these)