The automorphism tower problem revisited
  It is well-known that the automorphism towers of infinite centreless groups of cardinality kappa terminate in less than (2^{kappa})^+ steps. But an easy counting argument shows that (2^{kappa})^+ is not the best possible bound. However, in this paper, we will show that it is impossible to find an explicit better bound using ZFC. 
Winfried Just and Saharon Shelah and Simon Thomas