We show that for any countable group $ G $ equipped with a probability measure $ μ$, there exists a randomized stopping time $ τ$ such that $ (G, μ_τ )$ admits a strictly larger space of bounded harmonic functions than $ (G,μ) $, unless this space is trivial for all measures on $ G $. In particular, we exhibit an irreducible probability measure on the free group $F_2$ such that the Poisson boundary is strictly larger than the geometric boundary equipped with the hitting measure, resolving a longstanding open problem. As another consequence, there is never a nontrivial universal topological realization of the Poisson boundary for any countable group.