I apologize if the title (and the body) of this question is worded incorrectly, but I have no real experience in (transcendental) number theory, so it's probably the best I can do.
I've been thinking about irrational numbers recently, and a lot of my questions seem to come down to this: can we, in principle, show every transcendental number in some way?
Premise for the relevance of the question: Transcendentals must be shown rather than stated plainly, as can be done with (irrational) algebraic numbers. Perhaps I am wrong when I say this (please tell me if I am), but all definitions of π ultimately come down the ratio of a circle's circumference to it's diameter, and e must also be defined it terms of its use in a very specific, or rather, unique function. This is, more or less, what I mean by showing: that transcendental numbers must be given by their use in some definite structure ("structure" is here being used very broadly... again, I don't really have the proper mathematical vocabulary yet).
Now let me go back to my original question: can we always think of a context to fit any given transcendental number? Not necessarily a geometric construction, but some situation that provides a unique value that is a transcendental number.
Finally, links to any books, journal articles, etc., that may relate to or include this topic would be much appreciated.
I hope this made sense.