Are there mathematical objects that have been proved to exist but cannot be described in words?

You'd have to be more precise about what you mean by "describable in words". But you could argue that some irrational numbers can be described in words, like $\pi$ is the ratio of the circumference to the diameter of a circle. So if you consider each real number a mathematical object, then we could never possibly describe all of them because the set of things we can describe in language is necessarily countable, there are only a countable number of ways to assemble letters into words into sentences. But there are uncountably many real numbers. So inevitably most of them could never be described.

The reals are describable as a set, but you cannot describe each and every one of them in a way that distinguishes their individuality. Incidentally we can describe each and every element of $\mathbb N$ individually, because every natural number can be represented by a unique finite sequence of characters in the set $\{0,1,\dots,9\}$. The same cannot be said of $\mathbb R$.

Anything coming from the axiom of choice really. Such as the well ordering of the reals. Is there a known well ordering of the reals?

We can do it using set theory. The number of definable objects is countable, but the number of things that exist is uncountable. So something exists which isn't definable.

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Interestingly, you can't express definability in ZFC, in ZFC. If you try to, you get the following contradiction: https://mathoverflow.net/a/204794/1682.