Let $S$ be a surface (for simplicity, assume that $S$ has exactly one boundary component) and let $Mod(S)$ be its mapping class group. Let's assume that the genus of $S$ is at least $3$. To begin with, $Mod(S)$ is perfect, so its lower central series is not interesting. Define $\mathcal{I}(S)$ to be the Torelli group, ie the kernel of the action of $Mod(S)$ on $H_1(S)$. Things work better here. Johnson proved that the intersection of the lower central series for $\mathcal{I}(S)$ is trivial. Let me describe exactly what he did.
For a group $G$, let $\gamma_k(G)$ be the kth term in the lower central series for $G$, indexed so that $\gamma_0(G) = G$. Since $S$ has a boundary component, we can stick a basepoint $p$ on that boundary component and get an honest action of $Mod(S)$ on $\Gamma:=\pi_1(S,p)$ (if $S$ had no boundary, then we would only get an outer action). Johnson defined $\mathcal{I}(S,k)$ to be the kernel of the action of $Mod(S)$ on $\Gamma / \gamma_k(\Gamma)$. This gives a filtration $$\mathcal{I}(S) = \mathcal{I}(S,1) \supset \mathcal{I}(S,2) \supset \cdots$$ I believe that this is the filtration you are referring to (it has become known as the Johnson filtration). Johnson proved that $\cap_{k=1}^{\infty} \mathcal{I}(S,k) = 1$.
It is not hard to show that $\mathcal{I}(S,k) / \mathcal{I}(S,k+1)$ is abelian. One might thus be led to conjecture that $\mathcal{I}(S,k) = \gamma_{k-1}(\mathcal{I},k)$. Johnson proved that this is false. More specifically, he calculated $\mathcal{I}(S) / \gamma_1(\mathcal{I}(S))$ and showed that it contains a lot of 2-torsion coming from the Rochlin invariants of homology 3-spheres (the appropriate quotients were originally constructed by Birman and Craggs).
It is true that $\mathcal{I}(S,2)$ is the kernel of the universal torsion-free abelian quotient of $\mathcal{I}(S,1)$. One might thus conjecture that $\mathcal{I}(S,k)$ is the "torsion-free lower central series" of $\mathcal{I}(S)$. This hope was dashed by Morita, who showed that $\mathcal{I}(S,2)$ has $\mathbb{Z}$-quotients coming from the Casson invariants of homology 3-spheres that don't vanish on $\mathcal{I}(S,3)$.
We thus have two filtrations of $\mathcal{I}(S)$, the Johnson filtration and the "torsion-free lower central series". I'm not sure which is "better", but it is certainly true that the "torsion-free lower central series" is better understood due to a lot of work by Hain. In particular, he calculated a presentation for the Malcev completion of $\mathcal{I}(S)$, which is the completion of the filtered Lie algebra associated to the "torsion-free lower central series" filtration.