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Googolillion (or decitriotriacontillion) is an "-illionized" version of the googol, equal to $$10^{3\left(10^{100}\right)+3}$$ in the American system.[1] This number is also equal to 106*10100 in the long scale of naming zillions, accepted in French and Germany.

## Approximations

For short scale:

Notation Lower bound Upper bound
Arrow notation $$1000\uparrow(1+10\uparrow100)$$
Down-arrow notation $$50\downarrow\downarrow60$$ $$368\downarrow\downarrow40$$
Steinhaus-Moser Notation 56[3][3] 57[3][3]
Copy notation 2[2[101]] 3[3[101]]
H* function H(10H(32))
Taro's multivariable Ackermann function A(3,A(3,332)) A(3,A(3,333))
Pound-Star Notation #*((1))*(8,1,8,3)*10 #*((1))*(8,5,3,5,4)*7
BEAF {1000,1+{10,100}}
Hyper-E notation E(3+3E100)
Bashicu matrix system (0)(1)[331] (0)(1)[332]
Hyperfactorial array notation (69!)! (70!)!
Fast-growing hierarchy $$f_2(f_2(327))$$ $$f_2(f_2(328))$$
Hardy hierarchy $$H_{\omega^22}(327)$$ $$H_{\omega^22}(328)$$
Slow-growing hierarchy $$g_{\omega^{\omega^{\omega^2}3+3}}(10)$$

For long scale:

Notation Lower bound Upper bound
Arrow notation $$(10\uparrow6)\uparrow(10\uparrow100)$$
Down-arrow notation $$735\downarrow\downarrow36$$ $$892\downarrow\downarrow35$$
Steinhaus-Moser Notation 56[3][3] 57[3][3]
Copy notation 5[5[101]] 6[6[101]]
H* function H(19H(32)) H(20H(32))
Taro's multivariable Ackermann function A(3,A(3,333)) A(3,A(3,334))
Pound-Star Notation #*((1))*(5,1,4,6,4)*7 #*((1))*(11,10,6,1)*11
BEAF {{10,6},{10,100}}
Hyper-E notation E(6E100)
Bashicu matrix system (0)(1)[332] (0)(1)[333]
Hyperfactorial array notation (69!)! (70!)!
Fast-growing hierarchy $$f_2(f_2(328))$$ $$f_2(f_2(329))$$
Hardy hierarchy $$H_{\omega^22}(328)$$ $$H_{\omega^22}(329)$$
Slow-growing hierarchy $$g_{\omega^{\omega^{\omega^2}6}}(10)$$

1. Googology