Given a field, a polynomial of degree greater than 1, and a transcendental element over the ground field, we can visualize the (assumed to be disjoint) iterated preimages of the polynomial over this element as the vertices of a tree graph rooted at the transcendental element. The permutations of the elements in these sets by the appropriate Galois group correspond to automorphisms of the tree graph. Replacing the transcendental element with one in the ground field, we naturally obtain a subgroup of the original Galois group. If the ground field is Hilbertian, then every multivariate, irreducible polynomial over that field remains irreducible when we specialize some of the variables to become fixed values in the ground field outside a “thin set”, the algebraic analogue of “almost every”. One might guess this leads to the pro finite iterated Galois group over such ground field elements to have finite index in the same group structure over a transcendental element . To the contrary, every countable Hilbertian field has a countable Hilbertian extension such that the index between these pro finite Galois groups is infinite whenever the iterated Galois group over the transcendental element has infinitely many maximal normal subgroups. This is a strong generalization of a related result by Dittmann and Kadets.