This fact is used in a nice way by [Dunfield and Thurston][1] to show that, for any finite simple group Q, the number of Q-covers of a "random" 3-manifold in their sense follows a Poisson distribution.  (The multiple transitivity appears in Thm 7.4.)

Also:  I don't have a reference for this in mind, but I've seen Nick Katz give talks where he uses a "linear" version of this, showing that an algebraic subgroup of GL(V) (typically a monodromy group) is "as big as you expect", using irreducibility of tensor powers of V.

  [1]: http://arxiv.org/abs/math/0502567