Conjecturally, every finite group is the Galois group of some extension of the rationals. 
[This](https://mathoverflow.net/questions/53530/maximal-non-abelian-extensions-of-number-fields-unramified-everywhere) question made me wonder what is known about infinite
simple groups occurring as Galois groups. 


*What are the infinite simple groups that are expected to be Galois groups, i.e., profinite?* Are they classified? Are there any examples of such extensions?



Infinite simple Galois groups
Any profinite simple group is finite, since it has nontrivial finite quotients (the conjugates of a finite index subgroup intersect in a finite index subgroup).


