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I found a function that seems to be computable, but it is probably grows faster than any other computable.

I define s(n) by following rules:

We have basically decimal digits, indexes, # symbol means rest of expression, parenthesis, commas, the triple point (one symbol) and all elementary functions. Then we construct the expression, which can lead to a large number, using at most n symbols, and replace all variables to 10. We need to sort out all such possible expressions and form number from each one. The largest number is the value of s(n).

As n becomes larger, more notations might be defined, so s(n) stands above all recursive notations.

Some values:

\(s(0) = 0\), no possible expressions.

\(s(1) = 10\), one possible expression is single n

\(s(2) = 10^{10}\), I use standard notation and 2 symbols.

\(s(3) = 10^{10^{10}}\) = trialogue.

\(s(4) = 10^{10^{10^{10}}}\) = tetralogue.

\(s(5) = 10^{10^{10^{10^{10}}}}\) = pentalogue.

Then I define new 6 symbol notation: \({}^nn = n^{...^{n}}\) (there are 6 symbols, two n's before equality sign, third =, and n, ... and n for 4,5 and 6 symbol respectively. At this point, I am not sure that it is lead to the largest number defined using 6 symbols, but though it is lower bound, thus:

\(s(6) >= {}^{10}10 = 10 \uparrow \uparrow 10 \)

\(s(7) >= {}^{^{10}10}10 = 10 \uparrow \uparrow \uparrow 3 \)

\(s(8) >= {}^{^{^{10}10}10}10 = 10 \uparrow \uparrow \uparrow 3 \)

To explain what it is really powerful, I counted number of symbols which need to define multi-dimensional array notation might be defined with less than 335 symbols, thus:

\(s(335) > \lbrace 10,10 (10) 2 \rbrace \) = dimendecal.

Just imagine how big might be s(1000). s(1000000) seems to be surpass any number which ever defined by notation, s(googol) probably around Rayo's number.

What about s(s(5)). We need a pentalogue symbols to define this number. Nextly comes things like s(s(s(1))), s(s(s(s(...(s(s(1)...) (with s(s(1)) symbols), and so on.

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