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Here are some functions and groups of googolisms that I coined:

Groups

The Pies

Micropillion: 10^-(3•π+3)
Pillion: 10^3•π+3
Pienpi: π^π
Piendunpi: ^ππ
Pientrinpi: _ππ
Pienquadrinpi: π_π
Pibang: π!
Piquestione: π? (termial)
Pidubang, (π!)!
Piduquestione: (π?)?
Taubang: τ!
Tauquestione: τ? (termial)
Salada-pi-pah: (π! ^^^^ π! ^^ π!) ^^^^ π!
Tauillion: 10^3•τ+3
Piduoexpopi: π^π
Pitrioexpopi: π ^ π ^ π
Piquadrioexpopi: π ^ π ^ π ^ π
Pitriquestione: ((π?)?)?
Pitribang: ((π!)!)!
Pisald: π! ^^ π? ^^^ (π!)! ^^ π?
Pidusald: Pisald ^^^^^^ Pisald ^^ Pisalad
Pin of Pies: (((((π+1)x2)^ 3)^^ 4)^^^ 5
Pinterrobang: π!? (see list of googolisms)
Piduointerrobang: (π!?)!?
Root of Pies: π-root(π)
Piexclaquestione: (π!)?
Piquestiexcla: (π?)!
No pie for you!: π^-π
Superpi: ((π^ π)^^ π)^^^ π

Numbers using the H* function

Negatiyillion: H*(-1)
Zeryillion: H*(0)
Monyillion: H*(1)
Dekyillion H*(10)
Googolyillion: H*(10¹⁰⁰)
Googolplexyillion: H*(H*(10¹⁰⁰))
Googolduplexyillion: H*(H*(H*(10¹⁰⁰)))
Googoltriplexyillion: H*(H*(H*(H*(10¹⁰⁰))))
Superior-Googolyillion: H*(10¹⁰⁰,1)
Illion-Superior-Googolyillion: H^{10^100+1}(10¹⁰⁰) where supscript means iteration
Illion-Illion-Superior-Googolyillion: H^{10^100+2}(10¹⁰⁰)
Illion-Illion-Illion-Super-Googolyillion: H^{10^100+3}(10¹⁰⁰)
Bisuperior-Googolyillion: H*(10¹⁰⁰,2)
Illion-Bisuperior-Googolyillion: H^{Superior-Googolyillion+1}(10¹⁰⁰)
Trisuperior-Googolyillion: H*(10¹⁰⁰,3)
Superior-Superior-Googolyillion: H*(10¹⁰⁰,10¹⁰⁰)
Bisuperior-Superior-Googolyillion: H*(10¹⁰⁰,10¹⁰⁰,2)
Superior-Superior-Superior-Googolyillion: H*(10¹⁰⁰,10¹⁰⁰,10¹⁰⁰)
Super-Superior-Googolyillion: H*₂(10¹⁰⁰,1)
Duper-Superior-Googolyillion: H*₂(10¹⁰⁰,2)
Super-Superior-Superior-Googolyillion: H*₂(10¹⁰⁰,10¹⁰⁰)
Super-Superior-Superior-Superior-Googolyillion: H*₂(10¹⁰⁰,10¹⁰⁰,10¹⁰⁰)
Duper-Superior-Superior-Googolyillion: H*₃(10¹⁰⁰,10¹⁰⁰)
Truper-Superior-Superior-Googolyillion: H*₄(10¹⁰⁰,10¹⁰⁰)
Super-Super-Superior-Superior-Googolyillion: H*_{H*1(10¹⁰⁰,10¹⁰⁰)}(10¹⁰⁰,10¹⁰⁰)
Duper-Super-Superior-Superior-Googolyillion: A^Googol(1) where A(n) = H*_n(10¹⁰⁰,10¹⁰⁰)
Truper-Super-Superior-Superior-Googolyillion: B^Googol(1) where B(n) = A^A^A^A^…^A(n) (n)… (n)(n) with n A’s
Quadruper-Super-Superior-Superior-Googolyillion: C^Googol(1) where C(n) = B^B^B^B^…^B(n) (n)… (n) (n) with n C’s
Gargoogolyillion: H*(10²⁰⁰)
Giggolyillion: H*(10^^100)

Numbers based on faxul

Superpickaxul: 200$ (Pickover definition)
Superslouffexul: 200$ (Sloane and Plouffe definition)
Superdanixul: 200$ (Daniel Corrêa definition)
Hyperxul: H(200) (Hyperfactorial function)
Mixaxul: 200* (mixed factorial)
Expostfaxul: expostfacto(200) using Expostfacto function
Toriaxul: T(200) using torian function
q-axul: [200]₂₀₀! using q-factorial function
Factorexaxul: 200\ using factorexation
Subaxul: !200 using subfactorial
Composixul: 200!/200# using primorial
Semiprimaxul: Semiprimorial of 200
Weakaxul: wf(200) (Weak factorial function)
Bouncaxul: 200Λ (Bouncing factorial function)

Extended numbers using the alpha series in the fast-growing hierarchy

Zeralum series
Kila-Zeralum: f₀(1,000)
Mega-Zeralum: f₀(1,000,000)
Giga-Zeralum: f₀(1,000,000,000)
Tera-Zeralum: f₀(1,000,000,000,000)
Peta-Zeralum: f₀(1,000,000,000,000,000)
Exa-Zeralum: f₀(10¹⁸)
Zetta-Zeralum: f₀(10²¹)
Yotta-Zeralum: f₀(10²⁴)
Ronna-Zeralum: f₀(10²⁷)
Quetta-Zeralum: f₀(10³⁰)
Unalum series
Kila-Unalum: f₁(1,000)
Mega-Unalum: f₁(1,000,000)
Giga-Unalum: f₁(1,000,000,000)
Tera-Unalum: f₁(1,000,000,000,000)
Peta-Unalum: f₁(1,000,000,000,000,000)
Exa-Unalum: f₁(10¹⁸)
Zetta-Unalum: f₁(10²¹)
Yotta-Unalum: f₁(10²⁴)
Ronna-Unalum: f₁(10²⁷)
Quetta-Unalum: f₁(10³⁰)
Unalumplex: f₁(f₁(10))
Unalumduplex: f₁³(10) (superscript means iteration)
Unalumtriplex: f₁⁴(10)
Unalumtetraplex: f₁⁵(10)
Unalumpentaplex: f₁⁶(10)
Unalumhexaplex: f₁⁷(10)
Unalumheptaplex: f₁⁸(10)
Unalumoctaplex: f₁⁹(10)
Unalumennaplex: f₁¹⁰(10) = f₂(10) or Balum
Unalumdekaplex: f₁¹¹(10)
Unalumhectaplex: f₁¹⁰¹(10)
Unalumkilaplex: f₁^1,001(10)
Unalumegaplex: f₁^1,000,001(10)
Unalumgigaplex: f₁^{1,000,000,001}(10)
Unalumteraplex: f₁^{1,000,000,000,001}(10)
Unalumpetaplex: f₁^{1,000,000,000,000,001}(10)
Unalumexaplex: f₁^{10^18+1}(10)
Unalumzettaplex: f₁^{10^21+1}(10)
Unalumyottaplex: f₁^{10^24+1}(10)
Unalumronnaplex: f₁^{10^27+1}(10)
Unalumquettaplex: f₁^{10^30+1}(10)
Unalumdex: f₁^f₁(10)(10)
Balum series
Kila-Balum: f₂(1,000)
Mega-Balum: f₂(1,000,000)
Giga-Balum: f₂(1,000,000,000)
Tera-Balum: f₂(1,000,000,000,000)
Peta-Balum: f₂(1,000,000,000,000,000)
Exa-Balum: f₂(10¹⁸)
Zetta-Balum: f₂(10²¹)
Yotta-Balum: f₂(10²⁴)
Ronna-Balum: f₂(10²⁷)
Quetta-Balum: f₂(10³⁰)
Balumplex: f₂(f₂(10))
Balumduplex: f₂³(10)
Balumtriplex: f₂⁴(10)
Balumtetraplex: f₂⁵(10)
Balumpentaplex: f₂⁶(10)
Balumhexaplex: f₂⁷(10)
Balumheptaplex: f₂⁸(10)
Balumoctaplex: f₂⁹(10)
Balumennaplex: f₂¹⁰(10) = f₃(10) or Tralum
Balumdekaplex: f₂¹¹(10)
Balumhectaplex: f₂¹⁰¹(10)
Balumkilaplex: f₂^1,001(10)
Balumegaplex: f₂^1,000,001(10)
Balumgigaplex: f₂^{1,000,000,001}(10)
Balumteraplex: f₂^{1,000,000,000,001}(10)
Balumpetaplex: f₂^{1,000,000,000,000,001}(10)
Balumexaplex: f₂^{10^18+1}(10)
Balumzettaplex: f₂^{10^21+1}(10)
Balumyottaplex: f₂^{10^24+1}(10)
Balumronnaplex: f₂^{10^27+1}(10)
Balumquettaplex: f₂^{10^30+1}(10)
Balumdex: f₂^f₂(10)(10)
Tralum series
Kila-Tralum: f₃(1,000)
Mega-Tralum: f₃(1,000,000)
Giga-Tralum: f₃(1,000,000,000)
Tera-Tralum: f₃(1,000,000,000,000)
Peta-Tralum: f₃(1,000,000,000,000,000)
Exa-Tralum: f₃(10¹⁸)
Zetta-Tralum: f₃(10²¹)
Yotta-Tralum: f₃(10²⁴)
Ronna-Tralum: f₃(10²⁷)
Quetta-Tralum: f₃(10³⁰)
Tralumplex: f₃(f₃(10))
Tralumduplex: f₃³(10)
Tralumtriplex: f₃⁴(10)
Tralumtetraplex: f₃⁵(10)
Tralumpentaplex: f₃⁶(10)
Tralumhexaplex: f₃⁷(10)
Tralumheptaplex: f₃⁸(10)
Tralumoctaplex: f₃⁹(10)
Tralumennaplex: f₃¹⁰(10) = f₄(10) or Quadralum
Tralumdekaplex: f₃¹¹(10)
Tralumhectaplex: f₃¹⁰¹(10)
Tralumkilaplex: f₃^1,001(10)
Tralumegaplex: f₃^1,000,001(10)
Tralumgigaplex: f₃^{1,000,000,001}(10)
Tralumteraplex: f₃^{1,000,000,000,001}(10)
Tralumpetaplex: f₃^{1,000,000,000,000,001}(10)
Tralumexaplex: f₃^{10^18+1}(10)
Tralumzettaplex: f₃^{10^21+1}(10)
Tralumyottaplex: f₃^{10^24+1}(10)
Tralumronnaplex: f₃^{10^27+1}(10)
Tralumquettaplex: f₃^{10^30+1}(10)
Tralumdex: f₃^f₃(10)(10)
Omegalum series
Omegalum: f_ω(10)
Hecta-Omegalum: f_ω(100)
Kila-Omegalum: f_ω(1,000)
Mega-Omegalum: f_ω(1,000,000)
Giga-Omegalum: f_ω(1,000,000,000)
Tera-Omegalum: f_ω(1,000,000,000,000)
Peta-Omegalum: f_ω(1,000,000,000,000,000)
Exa-Omegalum: f_ω(10¹⁸)
Zetta-Omegalum: f_ω(10²¹)
Yotta-Omegalum: f_ω(10²⁴)
Ronna-Omegalum: f_ω(10²⁷)
Quetta-Omegalum: f_ω(10³⁰)
Omegalumplex: f_ω(f_ω(10))
Omegalumduplex: f_ω³(10)
Omegalumtriplex: f_ω⁴(10)
Omegalumtetraplex: f_ω⁵(10)
Omegalumpentaplex: f_ω⁶(10)
Omegalumhexaplex: f_ω⁷(10)
Omegalumheptaplex: f_ω⁸(10)
Omegalumoctaplex: f_ω⁹(10)
Omegalumennaplex: f_ω¹⁰(10) = f_ω+1(10)
Omegalumdekaplex: f_ω¹¹(10)
Omegalumhectaplex: f_ω¹⁰¹(10)
Omegalumkilaplex: f_ω^1,001(10)
Omegalumegaplex: f_ω^1,000,001(10)
Omegalumgigaplex: f_ω^{1,000,000,001}(10)
Omegalumteraplex: f_ω^{1,000,000,000,001}(10)
Omegalumpetaplex: f_ω^{1,000,000,000,000,001}(10)
Omegalumexaplex: f_ω^{10^18+1}(10)
Omegalumzettaplex: f_ω^{10^21+1}(10)
Omegalumyottaplex: f_ω^{10^24+1}(10)
Omegalumronnaplex: f_ω^{10^27+1}(10)
Omegalumquettaplex: f_ω^{10^30+1}(10)
Omegalumdex: f_ω^f_ω(10)(10)

Hardy hierarchy groups

Beta series
Zerahar: H₀(10)
Unohar: H₁(10)
Bihar: H₂(10)
Trihar: H₃(10)
Quadrahar: H₄(10)
Quintahar: H₅(10)
Sextahar: H₆(10)
Septahar: H₇(10)
Octahar: H₈(10)
Nonahar: H₉(10)
Dekahar: H₁₀(10)
Hectahar: H₁₀₀(10)
Kilahar: H_1,000(10)
Megahar: H_1,000,000(10)
Gigahar: H_{1,000,000,000}(10)
Terahar: H_{1,000,000,000,000}(10)
Petahar: H_{1,000,000,000,000,000}(10)
Exahar: H_10^18(10)
Zettahar: H_10^21(10)
Yottahar: H_10^24(10)
Ronnahar: H_10^27(10)
Quettahar: H_10^30(10)
Amon series
Amon: H_ω(10)
Unamon: H_ω+1(10)
Biamon: H_ω+2(10)
Triamon: H_ω+3(10)
Quadramon: H_ω+4(10)
Quintamon: H_ω+5(10)
Sextamon: H_ω+6(10)
Septamon: H_ω+7(10)
Octamon: H_ω+8(10)
Nonamon: H_ω+9(10)
Dekamon: H_ω+10(10)
Hectamon: H_ω+100(10)
Kilamon: H_ω+1,000(10)
Megamon: H_ω+1,000,000(10)
Gigamon: H_{ω+1,000,000,000}(10)
Teramon: H_{ω+1,000,000,000,000}(10)
Petamon: H_{ω+1,000,000,000,000,000}(10)
Exammon: H_ω+10^18(10)
Zettamon: H_ω+10^21(10)
Yottamon: H_ω+10^24(10)
Ronnamon: H_ω+10^27(10)
Quettamon: H_ω+10^30(10)
Bammon series
Bammon: H_ω2(10)
Unabammon: H_ω2+1(10)
Biabammon: H_ω2+2(10)
Triabammon: H_ω2+3(10)
Quadrabammon: H_ω2+4(10)
Quintabammon: H_ω2+5(10)
Sextabammon: H_ω2+6(10)
Septabammon: H_ω2+7(10)
Octabammon: H_ω2+8(10)
Nonabammon: H_ω2+9(10)
Dekabammon: H_ω2+10(10)
Hectabammon: H_ω2+100(10)
Kilobammon: H_ω2+1,000(10)
Megabammon: H_{ω2+1,000,000}(10)
Gigabammon: H_{ω2+1,000,000,000}(10)
Terabammon: H_{ω2+1,000,000,000,000}(10)
Petabammon: H_{ω2+1,000,000,000,000,000}(10)
Exabammon: H_ω2+10^18(10)
Zettabammon: H_ω2+10^21(10)
Yottabammon: H_ω2+10^24(10)
Ronnabammon: H_ω2+10^27(10)
Quettabammon: H_ω2+10^30(10)
Trammon series
Trammon: H_ω3(10)
Unatrammon: H_ω3+1(10)
Biatrammon: H_ω3+2(10)
Triatrammon: H_ω3+3(10)
Quadratrammon: H_ω3+4(10)
Quintatrammon: H_ω3+5(10)
Sextatrammon: H_ω3+6(10)
Septatrammon: H_ω3+7(10)
Octatrammon: H_ω3+8(10)
Nonatrammon: H_ω3+9(10)
Deketrammon: H_ω3+10(10)
Hectatrammon: H_ω3+100(10)
Kilotrammon: H_ω3+1,000(10)
Megatrammon: H_{ω3+1,000,000}(10)
Gigatrammon: H_{ω3+1,000,000,000}(10)
Teratrammon: H_{ω3+1,000,000,000,000}(10)
Petatrammon: H_{ω3+1,000,000,000,000,000}(10)
Exatrammon: H_ω3+10^18(10)
Zettatrammon: H_ω3+10^21(10)
Yottatrammon: H_ω3+10^24(10)
Ronnatrammon: H_ω3+10^27(10)
Quettatrammon: H_ω3+10^30(10)
n-mon series
Quadramon: H_ω4(10)
Quintamon: H_ω5(10)
Sextamon: H_ω6(10)
Septamon: H_ω7(10)
Octamon: H_ω8(10)
Nonamon: H_ω9(10)
Dekamon: H_ω10(10)
Hectamon: H_ω100(10)
Kilamon: H_ω1,000(10)
Megamon: H_ω1,000,000(10)
Gigamon: H_{ω1,000,000,000}(10)
Teramon: H_{ω1,000,000,000,000}(10)
Petamon: H_{ω1,000,000,000,000,000}(10)
Exammon: H_ω10^18(10)
Zettamon: H_ω10^21(10)
Yottamon: H_ω10^24(10)
Ronnamon: H_ω10^27(10)
Quettamon: H_ω10^30(10)From here on, it mimics the fast-growing hierarchy (H_ω^α(n) = f_α(n)) so it would be useless to continue further.

Numbers based on Graham’s number (g(n) means the Graham’s function)

Grandham: g₆₅
Granderham: g₆₆
Gloomy graham: a₆₄ where a₀ = 4 and a_n = 3v^a_n-13 where v means down-arrow
Smalham: g₆₃
Smalerham: g₆₂
Saladaham: Sam_{The least stable ordinal}(Σ^TREE[999!](TREE[g(g(g(g(64!^G!))))]) using nested factorial notation
Croutoham: g_C where C is crouto’s value on the final step

BIGGer than the BIGG

Microgrand-BIGG: (…(200?)?)…)? with 200 ?’s
Minigrand-BIGG: (…(200?)?)…)? with 201 ?’s
Grand-BIGG: (…(200?)?)…)? with BIGG+1 ?’s
Grand Kilo-BIGG: 200(?^Kilo-BIGG+1) (works exactly like nested factorial notation with ?’s)
Bigrand BIGG: 200(?^{Grand BIGG+1})
Bigrand Kilo-BIGG: 200(?^{Grand Kilo-BIGG+1})
Trigrand BIGG: 200(?^{Bigrand BIGG+1})
Quadgrand BIGG: 200(?^{Trigrand BIGG+1})
Quintgrand BIGG: 200(?^{Quadgrand BIGG+1})
BIGGgrand BIGG: A(200?), where A(n+1) = 200(?^A(n)+1) and A(0) = BIGG
BIGGestgrand BIGG: A^200?+1(200?)
Ultra Massive Big Hyper Power Ultramassive White Stupendous Strong Mega Power Energy System Mega Fuga Massive Eyes Salad Fish Garden Ridiculous Supermassive Blackhole Collapse Googology Exactly Blackhole Stupendously Hard Giga Fuga Tera Peta Exa Zetta Yotta Quetta Kilo Mega Hecta Deca Myriad Horrendous Stupendous Horrible Terrible Ohmygosh Thoth Tetration Petation Ultration Aperiogation Legislation In Cuneiform Ridiculous Beaf That Can’t Be Undone By Legends Master Ultra Gold Diamond Emerald Sapphire Crystal Moon Earth Sun Betelgeuse Powered Ultra BEAF Surrender Golden Diamond Ultrapowered By The BIGG:The Biggest Possible Number That Can Be Described In At Most BIGG Symbols Avoiding Every Paradox And Loops In ExistenceUltra Massive Big Hyper Power Ultramassive White Stupendous Strong Mega Power Energy System Mega Fuga Massive Eyes Salad Fish Garden Ridiculous Supermassive Blackhole Collapse Googology Exactly Blackhole Stupendously Hard Giga Fuga Tera Peta Exa Zetta Yotta Quetta Kilo Mega Hecta Deca Myriad Horrendous Stupendous Horrible Terrible Ohmygosh Thoth Tetration Petation Ultration Aperiogation Legislation In Cuneiform Ridiculous Beaf That Can’t Be Undone By Legends Master Ultra Gold Diamond Emerald Sapphire Crystal Moon Earth Sun Betelgeuse Powered Ultra BEAF Surrender Golden Diamond Ultrapowered By The BIGGer:The Biggest Possible Number That Can Be Described In At Most (The last entry) Symbols Avoiding Every Paradox And Loops In ExistenceUltra Massive Big Hyper Power Ultramassive White Stupendous Strong Mega Power Energy System Mega Fuga Massive Eyes Salad Fish Garden Ridiculous Supermassive Blackhole Collapse Googology Exactly Blackhole Stupendously Hard Giga Fuga Tera Peta Exa Zetta Yotta Quetta Kilo Mega Hecta Deca Myriad Horrendous Stupendous Horrible Terrible Ohmygosh Thoth Tetration Petation Ultration Aperiogation Legislation In Cuneiform Ridiculous Beaf That Can’t Be Undone By Legends Master Ultra Gold Diamond Emerald Sapphire Crystal Moon Earth Sun Betelgeuse Powered Ultra BEAF Surrender Golden Diamond Ultrapowered By The BIGGest:The Biggest Possible Number That Can Be Described In At Most (The last entry) Symbols Avoiding Every Paradox And Loops In Existence

Functions

Lock-Linear function (updated) (No longer in work)

Take a valid array where all numbers are positive integers larger than 0. Follow the rules according to its arrays type (linear, multi-level, nested level, etc):Define ‘…’ as the rest of a array.
Solve the most nested array first.1. Linear arrays rules:1.1: If the array is empty:
[] = 1
1.2: If the last entry is equal to one:
[…,1] = […]
1.3: If there’s three or more entries and the last entry is larger than 1:
[…,a,b,c] = […,a,[…,a,b,c-1],c-1]
1.4: If there’s exactly two entries and the last one is bigger than one:
[a,b] = a^b
1.5: If there’s only one entry and the value is bigger than one:
[a] = a2. Multi-level arrays rules:% means the rest of the level array2.1: If the last entry of right array is one:
[%][…,1] = [%][…]2.2: If the array has three or more entries and the last entry is bigger than one:
[%][…,a,b,c] = [%][…,a,([%][…,a,b,c-1]),c-1]2.3: If there’s exactly two entries on the right side, b is larger than one and there’s only one entry on the level (left) side and that entry is bigger than one:
[n][a,b] = [n-1][a,a,a,a,a,…] with b a’s2.4: If the level is one and there’s 2 entries and the last of the two entries are bigger than one:
[1][a,b] = [a,a,a,a,…] with b a’s2.5: If the level has 2 entries and the last one is larger than one:
[a,b][…] = [a]([a,b-1][…])2.6: If the last entry of the level is equal to one:
[%,1][…] = [%][…]2.7: If the level has 3 or more entries and the last entry of the level is larger than one:
[%,a,b,c][…] = [%,a,([%,a,b,c-1][…]),c-1][…]2.8: If there’s only one entry on the right side and that entry is bigger than one:
[%][a] = a2.9: If there’s no entries on the right side:
[%][] = 13. Nested multi-level arrays rules:In a array like (((([A₁][A₂])[A₃])…)[A_k]) ([A_i] is a valid level linear array), follow these rules:3.1 Initial reduction:
Repeatedly apply only Rules 2.1–2.9 to [A₁][A₂] until no further application is possible,
resulting in [T₁],
where [T₁] is the fully reduced form of [A₁][A₂] under Rules 2.1–2.9.3.2: Iteration:
When n is bigger than one, follow this rule:3.2.1: Evaluate [T_n][A_n+1] by repeatedly applying Rules 2.1–2.9
until it reduces to [T_n+1] where [T_n] is the fully reduced form of [T_n-1][A_n] under rules 2.1-2.9. Continue this process until you consumed all [A_n]’s. Then, you switch to rule 1 for more information on what you do.

Ultrafactorial % function

TO BE ANNOUNCED

Non-integer extension to triangular numbers, uparrow and interrobang:

Up-arrow extension: a ↑^xb
a if a or b=1
a^b if x=1 or 0 ≤ b ≤ 1
a ↑^x-1 (a ↑^x (b-1)) if b and x > 1Triangular numbers extension: a?
a+(a-1)+(a-2)+(a-3)…+3+2+1 if a is a positive integer
(a•(a+1))/2 if a does not belong to positive integersInterrobang extension:
((a•(a+1))/2)•a!