Do unexpressible numbers exist? I just learned about the difference between transcendental numbers and irrational numbers (I guess I had been mis-educated into thinking they were the same thing) and it made me wonder if there is such a thing as an unexpressible number. Pi is apparently transcendental, but it can be described as the circumference of a circle divied by it's diameter. e can also be described. But our mathematical language has a countably infinite number of descriptions while the set of irrational numbers is uncountably infinite. This suggests to me that there are irrational numbers that can't have a description? Is this correct? I found a wikipedia page on Indescribable Cardinals but it was way over my head. Are they the answer to my question?

I believe the concept of computable/uncomputable numbers is what you're looking for. In less technical language than the Wikipedia page, a number is computable if we can make a computer generate the first $n$ digits of the number (it doesn't matter how long it takes as long as it will eventually finish) for any integer $n$. Via root-finding algorithms, all algebraic numbers are computable, as are $\pi,e$, etc. via infinite series. So are some less obvious examples like $\Gamma(a/b)$ for any $a,b$ and some other things. But, similarly to what you were saying with "descriptions," the number of such algorithms is countable, so the cardinality of computable numbers is also countable.

So, not only are there uncomputable numbers, but "almost every" number is uncomputable.