  For each natural number $n$, we consider the subgroup $\mathcal{R}_n$ of
Homeo$_+[0,1]$ made by the elements that are linear except for a subset whose
Cantor-Bendixson rank is less than or equal to $n$. These groups of generalized
piecewise-linear transformations yield an ascending chain of groups as we
increase $n$. We study how the notion of distorted element changes along this
chain. Our main result establishes that for each natural number $n$, there
exits an element that is undistorted of $\mathcal{R}_n$ yet distorted in
$\mathcal{R}_{n+1}$. Actually, such an element is explicitly constructed.
