User blog comment:Ikosarakt1/Symbols function/@comment-5529393-20121206121247

No computable function is larger than all other computable functions;  if s(n) is computable, then you could take t(n) = s^n(n) and that would be a faster growing computable function. Then you could define a long ordinal hierarchy with s as your base function, etc.

An interesting question is whether all recursive functions are representable using your notation. It's hard for me to tell since it looks like your notation is somewhat ambiguous;  for example, if there is more than one equal sign how do you know what defines the final number?

Anyway, if not all computable functions are representable than perhaps s(n) could be computable, and it could be very fast-growing, although slower than BB(n) of course. An interesting example is the 512 character program that won the "BIgnum Bakeoff" contest held about a decade ago, that diagonalized out of the Calculus of Constructions. Since the Calculus of Constructions can represent any function that is provably recursive in nth order arithmetic for any n, the associated function will be larger than any computable function that we've managed to come up with here.

If all computable functions are representable, then s(n) not be computable, and will grow at least as fast as BB(n). I don't think that it will match up with Rayo's number, though - the language of set theory is extremely strong, stronger I think then the simple notation you've come up with here. So "the largest number definable in first order set theory using at most a googol symbols" should be bigger.