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Googolplexian, or more systematically, googolduplex, is a number equal to $$10^{10^{10^{100}}}$$, or 1 followed by a googolplex zeroes.[1][2] In Hyper-E notation, it is equal to E100#3. It is 1010100+1 digits long.

The number has many alternative names such as googolplexplex,[2] googolplusplex,[3] gargoogolplex (by Saibian),[4] googolplex2,[5] googolbiplex,[6] and most famously googolplexian.[2][7]

Writing down the full decimal expansion would take 1010100-6 books of 400 pages each, with 2,500 digits on each page (except for the first, which would have 2,501).

Ten to the power of googolduplex, $$10^{10^{10^{10^{100}}}}$$, is called googoltriplex.

Etymology

After back-forming the "-plex" suffix from "googolplex," the term "googolplexplex" follows naturally as double application of the suffix to googol. Googologists shortened this to "duplex," from the Greek prefix for "two," indicating that the "plex" suffix should be applied twice. The names "googolplusplex", "gargoogolplex", and "googolplexian" do not appear to be of any logical etymology, as they appear to be cultural, or in popular culture, other than being various mutations of googolplex, independent from conventional naming schemes.

Names in -illion systems

Using Saibian's generalization of Bowers' -illions, it is approximately:

Names in other notations

DeepLineMadom calls the number troogolduplex, and is equal to 10[3]10[3]10[3]100 in DeepLineMadom's Array Notation[8]. It should not to be confused with the much larger Bowers' troogolduplex.

Computation

Googolduplex is $$f($$googol$$)$$ when: $$\left\{ \begin{array}{ll} f(n)=f(n-1)^{10} \\ f(0)=10 \end{array} \right.$$

Approximations in other notations

Notation Approximation
Up-arrow notation $$10 \uparrow 10 \uparrow 10 \uparrow 100$$ (exact)
Chained arrow notation $$10 \rightarrow (10 \rightarrow (10 \rightarrow 100))$$ (exact)
Hyper-E notation $$\textrm{E}100\#3$$ (exact)
Hyperfactorial array notation $$((69!)!)!$$
BEAF $$\{10,\{10,\{10,100\}\}\}$$ (exact)
Fast-growing hierarchy $$f_2^{3}(326)$$
Hardy hierarchy $$H_{\omega^23}(326)$$
Slow-growing hierarchy $$g_{\omega^{\omega^{\omega^{\omega^2}}}}(10)$$ (exact)
Steinhaus-Moser Notation $$57[3][3][3]$$