Let $F$ be a free group of finite rank $r>1$ and let $c\_1,c\_2,...$ be a full set of basic commutators for $F$. Upon completing the $k$-th step of the collection process for an arbitrary $g\in F$, one obtains a representation 


$g=c\_1^{n\_1}c\_2^{n\_2}\ldots c\_{t}^{n\_t}g\_k$,


where 


$c\_1,\ldots,c\_t$ are all basic commutators of weight less than $k$ ($t$ depending on $k$),


$n\_1,\ldots,n\_t$ are integers,


$g\_k\in \gamma\_k$, the $k$-th term of the lower central series for $F$.


**Question.** Is there always $g\in G$ such that the series $g\_1,g\_2,\ldots$ never terminates (i.e. has nonidentity terms $g\_i$ for arbitrarily large $i$)?



Is the collection process for commutators potentially infinite? 
Yes. I believe (ab)^3 or so, at least (ab)^n for high enough n, never terminates. As far as I recall, only (ab)^1 and (ab)^2 terminate. If you start to collect them, I think you'll see the proof.


