  We propose an all-loop expression for scattering amplitudes in planar N=4
super Yang-Mills theory in multi-Regge kinematics valid for all multiplicities,
all helicity configurations and arbitrary logarithmic accuracy. Our expression
is arrived at from comparing explicit perturbative results with general
expectations from the integrable structure of a closely related collinear
limit. A crucial ingredient of the analysis is an all-order extension for the
central emission vertex that we recently computed at next-to-leading
logarithmic accuracy. As an application, we use our all-order formula to prove
that all amplitudes in this theory in multi-Regge kinematics are single-valued
multiple polylogarithms of uniform transcendental weight.
