  The main goal of the paper is to find the {\it absolute maximum} of the width
of the separatrix chaotic layer as function of the frequency of the
time-periodic perturbation of a one-dimensional Hamiltonian system possessing a
separatrix, which is one of the major unsolved problems in the theory of
separatrix chaos. For a given small amplitude of the perturbation, the width is
shown to possess sharp peaks in the range from logarithmically small to
moderate frequencies. These peaks are universal, being the consequence of the
involvement of the nonlinear resonance dynamics into the separatrix chaotic
motion. Developing further the approach introduced in the recent paper by
Soskin et al. ({\it PRE} {\bf 77}, 036221 (2008)), we derive leading-order
asymptotic expressions for the shape of the low-frequency peaks. The maxima of
the peaks, including in particular the {\it absolute maximum} of the width, are
proportional to the perturbation amplitude times either a logarithmically large
factor or a numerical, still typically large, factor, depending on the type of
system. Thus, our theory predicts that the maximal width of the chaotic layer
may be much larger than that predicted by former theories. The theory is
verified in simulations. An application to the facilitation of global chaos
onset is discussed.
