We have found the limit ℒ_h = lim inf (log₂ T)^2/3 _{T → ∞} ∥W(T·)/(2T log₂ T)^1/2 - h∥ for a Wiener process W and a class of twice weakly differentiable functions h ∈ C[0, 1], thus solving the problem of the convergence rate in Chung's functional law for the so-called "slowest points". Our description is closely related to an interesting functional emerging from a large deviation problem for the Wiener process in a strip.