The height of the automorphism tower of a group
  For a group G with trivial center there is a natural embedding of G into its automorphism group, so we can look at the latter as an extension of the group. So an increasing continuous sequence of groups, the automorphism tower, is defined, the height is the ordinal where this becomes fixed, arriving to a complete group. We first show that for any kappa=kappa^{aleph_0} there is a group of cardinality kappa with height>kappa^+ (improving the lower bound). Second we show that for many such kappa there is a group of height > 2^kappa, so proving that the upper bound essentially cannot be improved. 
Saharon Shelah