User blog:Edwin Shade/How do you Rayo-name BB(n) ?

"Using Rayo's microlanguage one can construct set elements of which are so called instantaneous descriptions of a Turing machine, and from this it's just a small step to define  Busy Beaver function . "

This is the sentence to found in the article for Rayo's function. I saw examples given for descriptions of numbers 0, 1 and 2, but I didn't see any other examples for the natural numbers.

If I wanted to, how could I go about making uncomputable functions using only the allowed symbols in Rayo's function ?

Furthermore, how could I Rayo-name numbers like a googol or a centillion in a reasonable amount of space ?