## Higher-order legions Edit

These numbers can be defined or approximated only using higher-order analogues of legions, in which Bowers' dubbed them (from ascending order) lugions, lagions, ligions, etc. He also used the L notation to devise analogues of arbitrarily high order.

- Goshomity, {100,100 \\\\\\...\\\\\\ 2} (100 \'s)
- Good goshomity, {100,100 \\\\\\...\\\\\\ 2} (goshomity \'s)
- Great goshomity, {100,100 \\\\\\...\\\\\\ 2} (good goshomity \'s)
- Big bukuwaha, {L2,bukuwaha}
_{100,100} - Bongo bukuwaha, {L3,big bukuwaha}
_{100,100} - Quabinga bukuwaha, {L4,bongo bukuwaha}
_{100,100} - Meameamealokkapoowa, {L100,10}
_{10,10} - Meameamealokkabipoowa, {LL100,10}
_{10,10} - Meameamealokkatripoowa, {LLL100,10}
_{10,10} - Meameamealokkaquadripoowa, {LLLL100,10}
_{10,10} - Meameamealokkapoowa oompa, {LLL...LLL,10}10,10 (meameamealokkapoowa (array of) L's)

## large BEAF numbers Edit

They are the largest known computably large numbers devised. Numbers generated by Finite promise games and the Greedy clique sequence may also belong to this group.

- Loader's number, D
^{5}(99)

But, yes. There is some huge n value at which point {3,n,1,2} starts to be at par with Loader's number as Loader's number is fixed but n can be close to infinity.

The logic is simple, it is sure that {3,Loader's number,1,2} >> Loader's number; so n must be far smaller than Loader's number. Range for n: SCG(13) < n < Loader's number

In 2013, the busy beaver number became computable, because n must be only **a few millions** for FOST(n) = busy beaver number. This means, it only takes a few million symbols, which can be easily stored by a computer (source: What are some of the values of Rayo(n)).

## computable numbers way beyond Rayo's number Edit

TREE^{Rayo's number}(3) = TREE(TREE(...TREE(3)...)), where the total nesting depth of the formula is Rayo's number levels of the TREE function.

SSCG^{Rayo's number}(3) = SSCG(SSCG(...SSCG(3)...)), where the total nesting depth of the formula is Rayo's number levels of the SSCG function.

SCG^{Rayo's number}(13) = SCG(SCG(...SCG(13)...)), where the total nesting depth of the formula is Rayo's number levels of the SCG function.

D^{Rayo's number}(99) = D(D(D...D(D(99))...))), where the total nesting depth of the formula is Rayo's number levels of the D function.

TREE(n), SSCG(n), SCG(n) and D(n) are all 4 computable and with nestings this long, it is sure, that these 4 entries are larger than Rayo's number.

The TREE-nesting was originally used by Adam P. Goucher and Ying Zheng's claim, that the Rayo hierarchy is the only way to exceed Rayo's number is wrong. There is some huge d value at which point D^{d}(99) starts to be at par with Rayo's number, as Rayo's number is fixed as Rayo(10^{100}), but d can be close to infinity. SCG^{c}(13), SSCG^{b}(3) and TREE^{a}(3) can also be as big as Rayo's number, if c, b and a are large enough.

It is sure, that D^{Rayo's number}(99) >> SCG^{Rayo's number}(13) >> SSCG^{Rayo's number}(3) >> TREE^{Rayo's number}(3) >> Rayo's number. So, for "D^{d}(99) = SCG^{c}(13) = SSCG^{b}(3) = TREE^{a}(3) = Rayo's number", d, c, b and a must all 4 be far smaller than Rayo's number. Range for d, c, b and a: Loader's number < d < c < b < a < Rayo's number