Given that you used the word "transcendental" in describing inexhaustibility, <a href="http://en.wikipedia.org/wiki/Schanuel%27s_conjecture">Schanuel's conjecture</a> seems to be an obvious instance.  In effect, Schanuel's conjecture implies that numbers such as $e+\pi$ that have "no reason" to be algebraic are indeed not algebraic.  See also the conjectures of Kontsevich and Zagier about <a href="http://en.wikipedia.org/wiki/Ring_of_periods">periods</a>, which have a similar flavor.