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Legion's number of the second kind is equal to $$666!^{666!} \approx 10^{1.609941\times10^{1,596}}$$, where $$666!$$ represents the factorial of 666.[1]

## Approximations

Notation Lower bound Upper bound
Arrow notation $$770\uparrow338\uparrow631$$ $$601\uparrow409\uparrow611$$
Down-arrow notation $$237\downarrow\downarrow673$$ $$596\downarrow\downarrow576$$
Steinhaus-Moser Notation 576[3][3] 577[3][3]
Copy notation 1[1[1597]] 2[2[1597]]
H* function H(536H(530)) H(537H(530))
Taro's multivariable Ackermann function A(3,A(3,5301)) A(3,A(3,5302))
Pound-Star Notation #*((1))*((92))*17 #*((1))*((93))*17
BEAF {770,{338,631}} {601,{409,611}}
Hyper-E notation E[237]672#2 E[596]575#2
Bashicu matrix system (0)(1)[72] (0)(1)[73]
Hyperfactorial array notation (666!)! (667!)!
Fast-growing hierarchy $$f_2(f_2(5291))$$ $$f_2(f_2(5292))$$
Hardy hierarchy $$H_{\omega^22}(5291)$$ $$H_{\omega^22}(5292)$$
Slow-growing hierarchy $$g_{\omega^{\omega^{\omega^3+\omega^25+\omega9+6}}}(10)$$ $$g_{\omega^{\omega^{\omega^3+\omega^25+\omega9+6}2}}(10)$$