One can generalize the ascending and descending central series by transfinite induction, setting $G _{\alpha +1}=[G_\alpha, G]$ and $ G_\beta= \cap _{\alpha <\beta} G_\alpha $ (and analogously for the upper centers).
What is known about these series? Above all, if one of these series terminates, does the other series also terminate at the same ordinal ? Is there a characterization of the groups where it terminates?
A free group $F$ is residually nilpotent, that is, its lower central series has $$F_{\omega} = \bigcap_{n \in \Bbb{N}} F_{n} = \{ 1 \},$$ but if $F$ is non-abelian, then $F$ is centreless, so its upper central series never gets a chance.
Consider now the locally dihedral group $G = D(2^{\infty})$, which is the extension of $Z(2^{\infty})$ by an involution inverting each element. Now $G$ is hypercentral (its upper central series terminates at $G$ after $\omega + 1$ steps), but the lower central series stops at $Z(2^{\infty})$. (See this article.)