We study the minimal possible growth of harmonic functions on lamplighters. We find that (Z=2)∿Z has no sublinear harmonic functions, (Z=2)∿Z² has no sublogarithmic harmonic functions, and neither has the repeated wreath product (... (Z=2∿Z²)∿Z²)∿...∿Z². These results have implications on attempts to quantify the Derriennic- Kaimanovich-Vershik theorem.