Is there at least one irrational number with the property that it cannot be defined by a finite string of information? 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 &amp; y" where x &amp; y bound the "otherwise-non-finitely-describable z, and z can be shown to be the only such number between x &amp; 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.
Yes, there are such numbers.  Chaitin discusses them extensively, he has a popular book on the subject, Meta Math: The Quest for Omega.  They do not form continuous connected segments of the real line, because for example, the rationals are dense (that is, between any two real numbers there is at least one rational -- and in fact there are an infinite number of rationals).

However, these non-computable numbers are also dense in the real line.

You might want to take a look at the "computability theory", "definable number", and "analytical hierarchy" articles in Wikipedia.

Note that this doesn't pose as big a problem for the philosophy of physics as you might think:  This doesn't say that you cannot measure a physical constant to whatever precision you might wish.
I don't think the set of all such numbers is well-defined. Using particular words to mean one thing makes defining other things more difficult. Is the set of irrationals that can be defined in English the same as the set that can be defined in French or Chinese?