This is fairly random, but I thought I might as well write it down. In general it seems hard to say when a tower of Galois extensions is itself Galois, but sometimes one can…

- The maximality-based scenario in Keenan Kidwell’s answer shows the Hilbert (ray) (p-)class field tower is Galois. (You may find Lemmermeyer’s survey, or the “Background” section here helpful.)
- An unramified (at finite places) abelian extension of a quadratic number field is Galois (see also MathOverflow). When the abelian extension in question is cyclic of degree 4, this comes up while constructing the “4-part” of the (narrow) Hilbert 2-class field (see Theorem 2 of Lemmermeyer, “Construction of Hilbert 2-class fields”), going beyond the construction of the “2-part” from classical genus theory. I was reminded of this after seeing Exercise 18.10 of Ireland & Rosen (find an example of a non-abelian CM-field) during a Directed Reading Program meeting today (01/29/16), but it’s not too closely related.