Here is a googology problem/thought experiment/philosophy question.

Suppose we write a finite message consisting of some number of ones and zeroes. We send that message to some distant, alien race that has studied virtually everything imaginable in mathematics. They know nothing about what the message means other than that it represents an enormous real number. We share virtually nothing in common with the aliens (other than being in the same universe with the same physical laws), so we can't just write an ASCII message in English describing the number — the aliens don't know English. In fact, we can't even draw a picture of the number because the aliens don't have our numeral system.

The idea is that we have to encode number in a completely self-contained, culturally independent mathematical package. (We can assume that the aliens think more or less the way we do. Think of this as describing a giant number to a foreign mathematician who doesn't understand your language or writing system.)

So what's the largest number you can name in \(10^{100}\) bits?