There are computable numbers whose minimum expression requires more bits than the universe can hold, using any possible encoding the universe has sufficient bits to define.

Infinity is much bigger than any finite number. This includes even "all possible numbers reachable in all possible finite encodings of a number within a given size", as inconceivably large as that is. No matter how large that set is, somewhere, there is some maximum number that is the largest possible number you can specify under the conditions I've given here, and that number is 0% of the way along the number line, which means that 100% of all "computable numbers" are larger than than that number.

There are some uncomputable numbers that have been given names. We even have the first few digits of some of them. See for instance: https://en.wikipedia.org/wiki/Chaitin%27s_constant

What does it take to think of a number? We can certainly refer to particular uncomputable numbers - choose an enumeration of Turing machines, and let x be the real number whose binary expansion has 1 at the i-th place if the i-th Turing machine halts, and 0 otherwise.

But I think all languages is countable, so it should be a subset of the product of two countable sets, which should be countable?