  We show that in the regime when strong disorder is more relevant than field
quantization the superfluid--to--Bose-glass criticality of one-dimensional
bosons is preceded by the prolonged logarithmically slow classical-field
renormalization flow of the superfluid stiffness at mesoscopic scales. With the
system compressibility remaining constant, the quantum nature of the system
manifests itself only in the renormalization of dilute weak links. On the
insulating side, the flow ultimately reaches a value of the Luttinger parameter
at which the instanton--anti-instanton pairs start to proliferate, in
accordance with the universal quantum scenario. This happens first at
astronomic system sizes because of the suppressed instanton fugacity. We
illustrate our result by first-principles simulations.
