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So there are several numbers to define:

Googolplexian is 10^{10^{10^{100}}}

Googoldupleplexian (a) is Googolplexian \uparrow \uparrow Googolplexian

Googoltrupleplexian (b) is a \rightarrow a \rightarrow 3

Googolquadrupleplexian (c) is b \rightarrow b \rightarrow 4

Googolpentrupleplexian is c \rightarrow c \rightarrow 5


This could be extended to a function

Googolplexian is stage 1


Dupleplex(x) = \text{if }x = 1 \text{ then return Googolplexian else } 
Dupleplex(x-1) \rightarrow Dupleplex(x-1) \rightarrow x Approximation:

(((10^{10^{10^{100}}}\uparrow\uparrow10^{10^{10^{100}}})\rightarrow (10^{10^{10^{100}}}\uparrow\uparrow10^{10^{10^{100}}})\rightarrow3)\rightarrow ((10^{10^{10^{100}}}\uparrow\uparrow10^{10^{10^{100}}})\rightarrow (10^{10^{10^{100}}}\uparrow\uparrow10^{10^{10^{100}}})\rightarrow3)\rightarrow4)\rightarrow ((10^{10^{10^{100}}}\uparrow\uparrow10^{10^{10^{100}}})\rightarrow (10^{10^{10^{100}}}\uparrow\uparrow10^{10^{10^{100}}})\rightarrow3)\rightarrow ((10^{10^{10^{100}}}\uparrow\uparrow10^{10^{10^{100}}})\rightarrow (10^{10^{10^{100}}}\uparrow\uparrow10^{10^{10^{100}}})\rightarrow3)\rightarrow4)\rightarrow5


Pseudocode:

// Upper hyper operators
function hyper(a, b, n):
    if n = 1:
        return a + b
    result := a
    repeat b - 1 times:
        result := hyper(a, result, n - 1)
    return result

// Lower hyper operators
function hyper_lower(a, b, n):
    if n = 1:
        return a + b
    result := a
    repeat b - 1 times:
        result := hyper_lower(result, a, n - 1)
    return result


function dupleplex(a):

    if a = 1:

        return 10^10^10^100

    return dupleplex(a - 1)

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