  In math.AC/9608214 it was shown that fields of generalized power series
cannot admit an exponential function. In this paper, we construct fields of
generalized power series with bounded support which admit an exponential. We
give a natural definition of an exponential, which makes these fields into
models of real exponentiation. The method allows to construct for every kappa
regular uncountable cardinal, 2^{kappa} pairwise non-isomorphic models of real
exponentiation (of cardinality kappa), but all isomorphic as ordered fields.
Indeed, the 2^{kappa} exponentials constructed have pairwise distinct growth
rates. This method relies on constructing lexicographic chains with many
automorphisms.
