  We construct a Hardy field that contains Ilyashenko's class of germs at
infinity of almost regular functions as well as all log-exp-analytic germs. In
addition, each germ in this Hardy field is uniquely characterized by an
asymptotic expansion that is an LE-series as defined by van den Dries et al. As
these series generally have support of order type larger than that of the set
of natural numbers, the notion of asymptotic expansion itself needs to be
generalized.
