  We study the minimal possible growth of harmonic functions on lamplighters.
We find that $(\mathbb{Z}/2)\wr \mathbb{Z}$ has no sublinear harmonic
functions, $(\mathbb{Z}/2)\wr \mathbb{Z}^2$ has no sublogarithmic harmonic
functions, and neither has the repeated wreath product
$(\dotsb(\mathbb{Z}/2\wr\mathbb{Z}^2)\wr\mathbb{Z}^2)\wr\dotsb\wr\mathbb{Z}^2$.
These results have implications on attempts to quantify the
Derriennic-Kaimanovich-Vershik theorem.
