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Anything in the $n-1$st term of the lower central series of the pure braid group should work. E.g. for $n=3$ let $\sigma_1$ and $\sigma_2$ be the two standard generators. Then $\sigma_1\sigma_2\sigma_1^{-1}\sigma_2^{-1}$ is Brunnian in your sense. Or for $n=4$, take $[\sigma_1,\[\sigma_2,\sigma_3]]$.