TITLE: Maximal (non-abelian) extensions of number fields unramified everywhere
QUESTION [8 upvotes]: Hello!
Let $K$ be a number field. All abelian unramified extensions are contained in the Hilbert class field which is a finite extension 'maximal' with respect to this property. For general unramified extensions, is there a bound (depending on $K$) on the degree of an unramified extension over $K$? If so, does the compositum of all unramified extensions also have finite degree over $K$ in general?
Thanks for your attention!
ADDENDUM: as Hunter noticed the answer can be no even just even for solvable groups, when the field admits an infinite class field tower. But perhaps it is still interesting to study the question for extension having simple Galois group, and possibly their compositum. Is there anything known about this case?

REPLY [4 votes]: It is worth pointing out a recent paper by Manabu Ozaki in the Inventiones where he proves that 

Given any finite $p$-group $G$ ($p$
  being a prime number), there exists a
  number field $F$ such that the group
  of $F$-automorphisms of the maximal
  unramified $p$-extension of $F$ is
  isomorphic to $G$.

See his Theorem I.  The paper is also available on the arXiv.