Burnside groups of exponent $n$ are length $n$ unfree. If $n\ge 665$, odd, then the free Burnside group of exponent $n$ is not amenable (see Adian's book "The Burnside problem" or Olshanskii's book "Geometry of defining relations" or my book "Combinatorial algebra: syntax and semantics", Chapter 5).

Burnside groups of exponent $n$ are length $n$ unfree. If $n\ge 665$, odd, then the free Burnside group of exponent $n$ of rank 2 or more is not amenable (see Adian's book "The Burnside problem" or Olshanskii's book "Geometry of defining relations" or my book "Combinatorial algebra: syntax and semantics", Chapter 5).