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Megafuga- is a prefix used on a number n to indicate $$^nn$$ using tetration (i.e. n pentated to 2).[1][2] It was invented by Stephen Houben, who asked, when hearing about the fuga- prefix:

Since the ^ operation is not associative, i.e. $$\left(x^y\right)^z \not= x^{\left(y^z\right)}$$, this begs the question whether Fuga(3) means $$\left(3^3\right)^3 = 19,683$$ or $$3^{\left(3^3\right)} = 7,625,597,484,987$$. Probably the latter, since the goal is to get big numbers....

Alistair Cockburn has kept fuga- as the former, and named the latter "megafuga-".

In Hyper-E notation, this is equal to E[n]1#n, and in Poly-cell notation, this is [n][n]<1>.

The first five values of megafuga-x are 1, 4, 7,625,597,484,987, 108.0723*10153, and 10101.3357*102,184. Houben noted, using a computer, that megafuga(4) is about $$4^{1.34 \times 10^{154}}$$, somewhat larger than $$4^{10^{100}}$$, and concluded that "computing all [the] digits of megafuga(4) will never happen." The decimal expansion of that number begins and ends with: 23,610,226,714,597,313,206............36,860,456,095,261,392,896 However, the number is far too large to write out all of the digits.

For megafuga-n:

Notation Approximation
Arrow notation $$n\uparrow\uparrow n$$
BEAF $$\{n,n,2\}$$
Bird's array notation $$\{n,n,2\}$$
Steinhaus-Moser Notation $$n-1[4]$$ lower bound $$n[4]$$ upper bound
Chained arrow notation $$n\rightarrow n\rightarrow 2$$
Hyperfactorial array notation $$n!1$$ lower bound $$n+1!$$ upper bound
Strong array notation s(n,n,2)
Fast-growing hierarchy $$f_3(n)$$
Hardy hierarchy $$H_{\omega^3}(n)$$

## Sources

Googological affixes

Suffixes: -teen · -ty · -plex · -illion · -yllion · -exian · -chime · -toll · -gong · -bong · -throng · -illiob
Prefixes: gar- · fz- · fuga- · megafuga- · booga- · trooga- · googo- · googolple-
SI prefixes: deca- · hecto- · kilo- · mega- · giga- · tera- · peta- · exa- · zetta- · yotta-
Obsolete SI prefixes: myria-
Extended SI prefixes: 1.001 · 1.01 · 1.1 · 1.5 · 2 · 3 · 666 · 105 · 107 · 108 · 1010 · 1011 · 1013 · 1014 · 1016 · 1017 · 1019 · 1020 · 1022 · 1023 · 1025 · 1027 · 1028 · 1029 · 1030 · 1031 · 1032 · 1033 · 1034 · 1035 · 1036 · 1039 · 1042 · 1045 · 1048 · 1050 · 1051 · 1054 · 1057 · 1060 · 1063 · 1066 · 1069 · 1072 · 1075 · 1078 · 1080 · 1081 · 1084 · 1087 · 1090 · 10100 · 10120 · 10150 · 10180 · 10210 · 10240 · 10270 · 10300 · 10600 · 10900 · 101200 · 101500 · 101800 · 102100 · 102400 · 102700 · 103000 and higher

Gar-: garone · gartwo · garthree · garfour · garfive · garsix · garseven · gareight · garnine · garten · garhundred · gargoogol · gartrialogue · gargoogolplex · gargiggol
Fz-: fzone · fztwo · fzthree · fzfour · fzfive · fzsix · fzseven · fzeight · fznine · fzten · fztwenty · fzhundred · fzthousand · fzmillion · fzgoogol · fzmilliplexion · fzgoogolplex · fzgargoogolplex · fzgargantugoogolplex · fzgiggol
Fuga-: fugaone · fugatwo · fugathree · fugafour · fugafive · fugasix · fugaseven · fugaeight · fuganine · fugaten · fugaeleven · fugahundred · fugagoogol · fugagoogolplex · fugagargoogolplex · fugagargantugoogolplex · fugagiggol
Megafuga-: megafugaone · megafugatwo · megafugathree · megafugafour · megafugafive · megafugasix · megafugaseven · megafugaeight · megafuganine · megafugaten · megafugaeleven · megafugatwelve · megafugathirteen · megafugafourteen · megafugafifteen · megafugasixteen · megafugaseventeen · megafugaeighteen · megafuganineteen · megafugatwenty · megafugathirty · megafugaforty · megafugafifty · megafugasixty · megafugaseventy · megafugaeighty · megafuganinety · megafugahundred · megafugagoogol · megafugagoogolplex · megafugagargoogolplex · megafugagargantugoogolplex
Booga-: boogaone · boogatwo · boogathree · boogafour · boogafive · boogasix · boogaten boogahundred · boogagoogol · boogagoogolplex
Gag: gagone · gagtwo · gagthree · gagfour · gagfive · gagsix · gagseven · gageight · gagnine · gagten · gaggoogol · gaggoogolplex