  Biased random walks on supercritical Galton--Watson trees are introduced and
studied in depth by Lyons (1990) and Lyons, Pemantle and Peres (1996). We
investigate the slow regime, in which case the walks are known to possess an
exotic maximal displacement of order $(\log n)^3$ in the first $n$ steps. Our
main result is another --- and in some sense even more --- exotic property of
biased walks: the maximal potential energy of the biased walks is of order
$(\log n)^2$. More precisely, we prove that, upon the system's non-extinction,
the ratio between the maximal potential energy and $(\log n)^2$ converges
almost surely to $\frac12$, when $n$ goes to infinity.
