  We have studied the physical evolution kernels for nine non-singlet
observables in deep-inelastic scattering (DIS), semi-inclusive
e^+e^-annihilation and the Drell-Yan (DY) process, and for the flavour-singlet
case of the photon- and heavy-top Higgs-exchange structure functions (F_2,
F_phi) in DIS. All known contributions to these kernels show an only
single-logarithmic large-x enhancement at all powers of 1-x. Conjecturing that
this behaviour persists to (all) higher orders, we have predicted the highest
three (DY: two) double logarithms of the higher-order non-singlet coefficient
functions and of the four-loop singlet splitting functions. The
coefficient-function predictions canbe written as exponentiations of
1/N-suppressed contributions in Mellin-N space which, however, are less
predictive than the well-known exponentiation of the ln^k N terms.
