Ok, so maybe that wasn't the best way of phrasing the question, but I think it's specific enough. Let me explain myself a bit more below in case I am wrong.
So I'm assuming (although I've never checked) that the irrational numbers are defined as simply all reals that are not rational.
I'm asking about the existence, then, of an irrational number that has no finite description. I.e, not only does it not have a finite-number or recurring decimal description, but no other description could be made (counting things such as "The ratio of the circumference and diameter of any circle in a Euclidean plane" as finite).
Clearly, there would have to be infinitely many of these and they would have to form continuous connected regions of the real line, otherwise they would afford descriptions such as "The otherwise-non-finitely-describable number lying between x & y" where x & y bound the "otherwise-non-finitely-describable z, and z can be shown to be the only such number between x & y.
If so, then this would have implications for Laplace's Demon and other similar philosophical arguments since it would be mathematically impossible to have knowledge of all of the universe so long as at least one parameter lay within one of these non-finitely-describable regions.