Not only there are more real numbers than we can describe, there are more natural numbers than we can describe.
Our abilities to describe somethings, while virtually unlimited, are quite limited. We are bounded by time, by space, by the lexicon of our language. We are bound by mortality. It follows, if so, that the largest number that you can even begin to describe, is quite small (that is, almost all the other numbers are larger).
Taking this to a somewhat more formal level, if by "describe" you mean define somehow using a first-order formula in a language including addition, multiplication, elementary functions, integration, predicates for "known sets" (e.g. the integers, and so on), the familiar constants (e.g. $\pi$, $e$), and all those things that you know... you could still only describe only a very few numbers, i.e. a countable number of them. Since there are uncountably many real numbers, we have shown that almost all the real numbers cannot be described.
But even then there are caveats about what do we mean by the word "define". There are models of set theory in which every element is definable without parameters. Including every real number. There are subtle things to mind here, and if you are interested, then I suggest you to read (at least the well written introduction) the following paper:
J. D. Hamkins, D. Linetsky, and J. Reitz, “Pointwise definable models of set theory,” Journal of Symbolic Logic, vol. 78, iss. 1, pp. 139-156, 2013.
And the arXiv version can be found here as well.
Keep in mind, though, that some understanding in logic is required to even approach this question seriously. You should have a good grasp on basic things like the difference between theory and meta-theory.