Interestingly, this is the concept of an “unenumerable number” which in ancient and medieval India was investigated by Jain mathematicians. They considered it an intermediate category of number between finite and infinite.
In terms of modern mathematics, though, I don’t see why it would be theoretically impossible to come up with a notation to express any finite number. Maybe it could be shown that there are some numbers that are so big that no notation could express them except by using so many terms that they would be realistically impossible to state in any reasonable time period. But surely if a finite (rational) number exists, it is theoretically capable of being numerically expressed in some way or other. Surely?
In practical terms, of course, the expressibility of a large number depends on the flexibility of your number system. The ancient mathematician Archimedes wrote a book to demonstrate that a notation could be constructed that could express the number of grains of sand it would take to fill up the universe (of course, he thought the universe was considerably smaller than we now do). He had to put in some work to be able to express numbers going up even as far as ((10^8)^(10^8))^(10^8), which is fairly sizable but insignificant in comparison with something like Graham’s number.