What was the motivation behind Hilbert's 7th problem?
Looking into some of the history behind transcendental number theory, it seems that the field was almost non-existent in the late 1800's/early 1900's. There was the result of Liouville proving they existed, but not much else. So what led Hilbert to believe the problem was so critical? Did he have some far-reaching applications of a solution in mind? The Gel-fond Schneider theorem seems to have only one notable extension since it was proven in 1935, so (and I may be horribly mistaken) this makes me think that the problem is not too important.
Hilbert himself considered the problem out of reach for 19 century math. That is why he included it. He thought that any solution would bring completely new methods to mathematics. The theory of transcendental numbers, the approximation theory of numbers and related fields is a part of the mainstream of current mathematics.