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What about the ultimate five-year-old’s answer? “Googleplex raised to the googleplex raised to the googleplex raised to the… (until the voice wears out).” / We came up with ‘’fuga’’ (pronounced ‘few-ga’). ‘’Fuga’’ is a mixture of the musical word “fugue,” and Kieran’s “gar-” prefix.
  —Alistair Cockburn


Fuga- is a prefix devised by Allstair Cockburn and his children, including the then six-year-old Kieran Cockburn for the sole purpose of continuing the gar- prefix invented by his son as mentioned in his A Fuga Really Big Numbers[1] blog. It is the successor of the fz- prefix. Fuga- is a portmanteau of fugue and gar-, and it means “that number raised to that number that number of times,” as Cockburn considers how a five-year-old who recently learned about powers might say “Googleplex raised to the googleplex raised to the googleplex raised to the… (until the voice wears out)” to beat others in naming the larger number.

The intended expression for fuga-n is \(\underbrace{n ↑ n ↑ n ↑ ... ↑ n}_n\).

Since exponentiation is not power associaive, fuga- is ambiguous, as noticed by Stephan Houben. Fuga- is now used on a number \(n\) to indicate left-associative repeated exponentiation \(((...((n^n)^n)^n ... ^n)\) with n amount of n's, which is equivalent to both \(n^{n^{n-1}}\) and \(n\downarrow \downarrow n\) in down-arrow notation. The more familiar right-associative repeated exponentiation is now called the megafuga- prefix.

The first few values of fuga-n are 1, 4, 19,683, 3.4028*1038, 7.1821*10436, 8.0191*106050, and 8.6958*1099424. It is an example of a function with growth rates slightly above double-exponential.

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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- · ronna- · quetta-
Non-SI prefixes: 1.001 · 1.01 · 1.1 · 1.5 · 2 · 3 · 666 · 104 · 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 · garmillion · gargoogol · gareceton · gartrialogue · gargoogolplex · gargiggol
Fz-: fzone · fztwo · fzthree · fzfour · fzfive · fzsix · fzseven · fzeight · fznine · fzten · fztwenty · fzthirty · fzhundred · fzthousand · fzmillion · fzgoogol · fzmilliplexion · fzgoogolplex · fzgargoogolplex · fzgargantugoogolplex · fzgiggol
Fuga-: fugaone · fugatwo · fugathree · fugafour · fugafive · fugasix · fugaseven · fugaeight · fuganine · fugaten · fugahundred · fugagoogol · fugagoogolplex · fugagargoogolplex · fugagargantugoogolplex · fugagiggol
Megafuga-: megafugaone · megafugatwo · megafugathree · megafugafour · megafugafive · megafugasix · megafugaseven · megafugaeight · megafuganine · megafugaten · megafugahundred · megafugagoogol · megafugagoogolplex · megafugagargoogolplex · megafugagargantugoogolplex · megafugagrangol
Extensions:
Booga-:: boogaone · boogatwo · boogathree · boogafour · boogafive · boogasix · boogaten · boogahundred · boogagoogol · boogagoogolplex
Gag-:: gagone · gagtwo · gagthree · gagfour · gagfive · gagsix · gagseven · gageight · gagnine · gagten · gaggoogol · gaggoogolplex

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