Examples of central series which are not lower central series. In this paper by Richard D. Wade, the central series for an arbitrary group $G$ is defined (definition below). Proposition $2.1$ (again, below) then states that the lower central series for a group $G$ is always contained within a central series.
What are non-trivial examples, given a group $G$, of central series which are not lower central series?

Definition: Let $G$ be a group. Let $\{G_k\}_{k \geq 1}$ be a sequence of subgroups of $G$ such that for all $k,l$, 
  
  
*
  
*$G_1 = G$
  
*$G_{k+1} \leq G_k$
  
*$[G_k,G_l] \subseteq G_{k+l}$
  where $[G_k,G_l]$ denotes the commutator subgroup.
  
  
  Then $\{G_k\}_{k \geq 1}$ is a central series of $G$.

.

Proposition: Suppose that $\{\gamma_k\}_{k\geq 1}$ is the lower central series for $G$, and suppose that $\{G_k\}_{k \geq 1}$ is a central series. Then for all $k$, $\gamma_k \subseteq G_k$.

 A: Any series of subgroup in abelian group will be an example, but it is sort of boring. More interesting one is dimension series of a group. Let $\Delta_R G$ be the ideal in the group ring $R[G]$ which is a kernel of homomorphism $R[G] \to G, g \mapsto 1_R$ (it is just elements with sum of coefficients equal to $0$). Then one may note that intersection $G \cap 1 + \Delta^n$ is actually a subgroup, and the fact that $\Delta^n/\Delta^{n+1}$ is invariant under conjugation in $R[G]/\Delta^{n+1}$ (because $\Delta$ is a two-sided ideal) gives that this series of subgroups is central. They are usually denoted as $D_R^n(G)$ or just $D^n(G)$ when $R = \Bbb Z$.
A lot of people in 40-60s tried to prove that it coincides with l. c. series (and did so unsuccesfully — there were plenty of papers with wrong or incomplete proofs). Magnus (who invented it) proved that for free groups it is true. But in 1969 Eliyahu Rips concocted an example of a finite nilpotent group  of class $3$ with $D^4(G) = \Bbb Z/2$. This is actually sharp — $D^{\leq 3} = \gamma_{\leq 3}$ for all groups (for $n = 1, 2$ it is obvious, but for $3$ it is an interesting theorem). Now there are some, but not plenty, recipes for construction of groups with dimension and lower central series diverging. This divergence $D^n/\gamma_n$ is always torsion, and has universally bounded exponent by function like $n!^n$ (I do not remember clearly, but for a prime $p$ there is no $p$-torsion for $n < p$). 
