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The geepatoth is equal to $$100\uparrow\uparrow(4\times100)\ \&\ 10$$ using the array of operator and BEAF.[1] The term was coined by Cookie Fonster. Since BEAF has not been formalised by Bowers at this level, this number is currently ill-defined.

Approximations

Since it is not defined by now, the following approximation is just an expected value under climbing method (strong bound):

Notation Approximation
Fast-growing hierarchy $$f_{\varepsilon_{\varepsilon_{\varepsilon_{\varepsilon_0}}}}(100)$$ (debatable)
Hardy hierarchy $$H_{\varepsilon_{\varepsilon_{\varepsilon_{\varepsilon_0}}}} (100)$$ (debatable)

Sources

1. Pointless Gigantic List of Numbers - Part 5: The Higher GiantsPointless Large Number Stuff. Retrieved 2017-06-07.