  Weighted recursive trees are built by adding successively vertices with
predetermined weights to a tree: each new vertex is attached to a parent chosen
at random with probability proportional to its weight. In the case where the
total weight of the tree at step $n$ grows polynomially in $n$, we obtained in
(Pain-S\'enizergues 2022) an asymptotic expansion for the height of the tree,
which falls into the university class of the maximum of branching random walks.
In this paper, we consider the case of a total weight growing sub-polynomially
in $n$ and obtain asymptotics for the height of the tree in several regimes,
showing that universality is broken and exhibiting new behaviors.
