Definitions use a slightly unformalised setup which is relatable to other notations.
(1) goppatoth = {10,100(X^^X)2}
(2) goppatothplex = {10,{10,100(X^^X)2}(X^^X)2}
(3) goppatothduplex = {10,{10,{10,100(X^^X)2}(X^^X)2}(X^^X)2}
(4) goppatothtriplex = {10,{10,{10,{10,100(X^^X)2}(X^^X)2}(X^^X)2}(X^^X)2}
(5) goppatothquadriplex = {10,{10,{10,{10,{10,100(X^^X)2}(X^^X)2}(X^^X)2}(X^^X)2}(X^^X)2}
(6) goppatogiggol = {10,100,2(X^^X)2}
(7) goppatogaggol = {10,100,3(X^^X)2}
(8) goppatogeegol = {10,100,4(X^^X)2}
(9) goppatogigol = {10,100,5(X^^X)2}
(10) goppatogoggol = {10,100,6(X^^X)2}
(11) goppatogagol = {10,100,7(X^^X)2}
(12) goppatoboogol = {10,10,100(X^^X)2}
(13) goppatocorporal = {10,100,1,2(X^^X)2}
(14) goppatomulporal = {10,100,2,2(X^^X)2}
(15) goppatopowporal = {10,100,3,2(X^^X)2}
(16) goppatoterporal = {10,100,4,2(X^^X)2}
(17) goppatopenporal = {10,100,5,2(X^^X)2}
(18) goppatobiggol = {10,10,100,2(X^^X)2}
(19) goppatocorplodal = {10,100,1,3(X^^X)2}
(20) goppatomulplodal = {10,100,2,3(X^^X)2}
(21) goppatopowplodal = {10,100,3,3(X^^X)2}
(22) goppatoterplodal = {10,100,4,3(X^^X)2}
(23) goppatopenplodal = {10,100,5,3(X^^X)2}
(24) goppatobaggol = {10,10,100,3(X^^X)2}
(25) goppatobeegol = {10,10,100,4(X^^X)2}
(26) goppatobigol = {10,10,100,5(X^^X)2}
(27) goppatoboggol = {10,10,100,6(X^^X)2}
(28) goppatobagol = {10,10,100,7(X^^X)2}
(29) goppatotroogol = {10,10,10,100(X^^X)2}
(30) goppatotriggol = {10,10,10,100,2(X^^X)2}
(31) goppatotraggol = {10,10,10,100,3(X^^X)2}
(32) goppatotreegol = {10,10,10,100,4(X^^X)2}
(33) goppatotrigol = {10,10,10,100,5(X^^X)2}
(34) goppatotroggol = {10,10,10,100,6(X^^X)2}
(35) goppatotragol = {10,10,10,100,7(X^^X)2}
(36) goppatoquadroogol = {10,10,10,10,100(X^^X)2}
(37) goppatoquintoogol = {10,10,10,10,10,100(X^^X)2}
(38) goppatosextoogol = {10,10,10,10,10,10,100(X^^X)2}
(39) goppatoseptoogol = {10,10,10,10,10,10,10,100(X^^X)2}
(40) goppato-octoogol = {10,10,10,10,10,10,10,10,100(X^^X)2}
(41) goppatononoogol = {10,10,10,10,10,10,10,10,10,100(X^^X)2}
(42) goppatodecoogol = {10,10,10,10,10,10,10,10,10,10,100(X^^X)2}
(43) goppatogoobol = {10,100(1)2(X^^X)2}
(44) goppatogoxxol = {10,100(2)2(X^^X)2}
(45) goppatocoloxxol = {10,100(3)2(X^^X)2}
(46) goppatoteroxxol = {10,100(4)2(X^^X)2}
(47) goppatopetoxxol = {10,100(5)2(X^^X)2}
(48) goppato-ectoxxol = {10,100(6)2(X^^X)2}
(49) goppatogongulus = {10,10(100)2(X^^X)2}
(50) goppatogoplexulus = {10,100((1)1)2(X^^X)2}
(51) goppatogoduplexulus = {10,100((0,1)1)2(X^^X)2}
(52) goppatogotriplexulus = {10,100(((1)1)1)2(X^^X)2}
(53) goppatogoquadriplexulus = {10,100(((0,1)1)1)2(X^^X)2}
(54) goppatoth-mul-two = {10,100(X^^X)3}
(55) goppatoth-mul-three = {10,100(X^^X)4}
(56) goppatoth-mul-four = {10,100(X^^X)5}
(57) goppatoth-mul-boogol = {10,10(X^^X)100}
The difference between my -mul- and XRQ CORPORATION's -with- is that -mul- ALWAYS KEEPS EVERYTHING 10 APART FROM THE MOST SIGNIFICANT ENTRY, WHICH IS 100. A goppatoth-with-boogol is already defined as {10,100(X^^X)100} (XRQ's site uses ([1]1) instead of (X^^X))
(58) goppatoth-mul-troogol = {10,10(X^^X)10,100}
(59) goppatoth-mul-quadroogol = {10,10(X^^X)10,10,100}
(60) goppatoth-mul-goobol = {10,100(X^^X)(1)2}
(61) goppatoth-mul-gootrol = {10,100(X^^X)(1)3}
(62) goppatoth-mul-gooquadrol = {10,100(X^^X)(1)4}
(63) goppatoth-mul-gossol = {10,10(X^^X)(1)100}
(64) goppatoth-mul-goxxol = {10,100(X^^X)(2)2}
(65) goppatoth-mul-coloxxol = {10,100(X^^X)(3)2}
(66) goppatoth-mul-teroxxol = {10,100(X^^X)(4)2}
(67) goppatoth-mul-petoxxol = {10,100(X^^X)(5)2}
(68) goppatoth-mul-ectoxxol = {10,100(X^^X)(6)2}
(69) goppatoth-mul-gongulus = {10,10(X^^X)(100)2}
(70) goppatoth-mul-goplexulus = {10,100(X^^X)((1)1)2}
(71) goppatoth-mul-goduplexulus = {10,100(X^^X)((0,1)1)2}
(72) goppatoth-mul-gotriplexulus = {10,100(X^^X)(((1)1)1)2}
(73) goppatoth-mul-goquadriplexulus = {10,100(X^^X)(((0,1)1)1)2}
These numbers already have names given by XRQ Corporation.
I still have -pow-two, -pow-root, etc. suffixes but they are not used right now.
I'll use saibian's deutero, trito, teterto, etc.
(74) deutero-goppatoth = {10,100(X^^X)(X^^X)2}
(75) trito-goppatoth = {10,100(X^^X)(X^^X)(X^^X)2}
(76) teterto-goppatoth = {10,100(X^^X)(X^^X)(X^^X)(X^^X)2}
(77) pepto-goppatoth = X^^X * 4 + X^^100 & 10
(78) exto-goppatoth = X^^X * 5 + X^^100 & 10
(79) epto-goppatoth = X^^X * 6 + X^^100 & 10
(80) ogdo-goppatoth = X^^X * 7 + X^^100 & 10
(81) ento-goppatoth = X^^X * 8 + X^^100 & 10
(82) dekato-goppatoth = X^^X * 9 + X^^100 & 10
We can play copycat with saibian again.
(83) goppafact = X^^X * 100 & 10 = {10,100(X^^X * X)2}
(By the way, this googolism was originally coined by XRQ Corporation)
This is NOT equal to a hecato-goppatoth because that would be X^^X * 99 + X^^100 & 10 and we have X^^X * 100 & 10 which decomposes to X^^X * 99 + X^^10 & 10 so a hecato-goppatoth is slightly bigger.
Here let's think of stuff on our own.
Saibian's &1-ipso-&2 operator does &1^&2
We can use it! For the sake of convenience I'll use the notation XRQ uses too...
(84) goppatoth-ipso-goxxol = {10,100(X^^X * X^2)2} = {10,100(2[1]1)2}
(85) goppatoth-ipso-coloxxol = {10,100(X^^X * X^3)2} = {10,100(3[1]1)2}
(86) goppatoth-ipso-teroxxol = {10,100(X^^X * X^4)2} = {10,100(4[1]1)2}
(87) goppatoth-ipso-petoxxol = {10,100(X^^X * X^5)2} = {10,100(5[1]1)2}
(88) goppatoth-ipso-ectoxxol = {10,100(X^^X * X^6)2} = {10,100(6[1]1)2}
(89) goppatoth-ipso-gongulus = {10,100(X^^X * X^X)2} = {10,100(0,1[1]1)2}
(90) goppatoth-ipso-goplexulus = {10,100(X^^X * X^X^X)2} = {10,100((1)1[1]1)2}
(91) goppatoth-ipso-goduplexulus = {10,100(X^^X * X^^4)2} = {10,100((0,1)1[1]1)2}
(92) goppatoth-ipso-gotriplexulus = {10,100(X^^X * X^^5)2} = {10,100(((1)1)1[1]1)2}
(93) goppatoth-ipso-goquadriplexulus = {10,100(X^^X * X^^6)2} = {10,100(((0,1)1)1[1]1)2}
(94) goppaduliath = {10,100((X^^X)^2)2} = {10,100([1]2)2}
(95) goppathruliath = {10,100((X^^X)^3)2} = {10,100([1]3)2}
(96) goppaterliath = {10,100((X^^X)^4)2} = {10,100([1]4)2}
(97) goppapepliath = {10,100((X^^X)^5)2} = {10,100([1]5)2}
(98) goppa-exliath = {10,100((X^^X)^6)2} = {10,100([1]6)2}
(99) goppa-epliath = {10,100((X^^X)^7)2} = {10,100([1]7)2}
(100) goppa-ocliath = {10,100((X^^X)^8)2} = {10,100([1]8)2}
(101) goppa-enliath = {10,100((X^^X)^9)2} = {10,100([1]9)2}
(102) goppadekliath = {10,100((X^^X)^10)2} = {10,100([1]10)2}
Phew... Now we reached the great Monster-Giant.
In xE^, its definition is E100(#^^#)^(#^^#)^#100
And its brother, Brother-Giant is E100#^^#^#^^#^#100 without parentheses. It's much bigger.
For something in BEAF to compare it would be
To be continued.......