I've asked this question before on Mathematics, and they suggested me to ask here (Link).
Is there an example of a simple infinite $2$-group?
If a $2$-group is Artinian I know that it also locally finite, so the simple $2$-group cannot be Artinian.
Take the subgroup generated by the elements of order $2$, it must coincide with $G$. If we have a periodic subgroup generated by two elements of order $2$, like $\langle a,\, b\rangle$, it must be finite.