Anything in the $n-1$st term of the lower central series of the pure braid group should work. 
The pure braid group is generated by generators $\beta_{i,j}$ where the $i$th strand pushes a finger over the intervening strands and hooks with the $j$th strand.
Then a commutator $[\beta_{i,j},\beta_{\ell,m}]$ is Brunnian in your sense. If you want to make the commutator involve every strand of the braid in a significant way, you can take an iterated commutator of these basic generators. For example, for $n=4$ you can consider $[\beta_{1,2},[\beta_{1,3},\beta_{1,4}]]$.

Something like $[\sigma_1^2,[\sigma_2^2,\sigma_3^2]]$ would also work.