List of googologisms/Higher computable level

  List of googologisms
 * Class 0 and 1
 * Class 2
 * Class 3
 * Class 4
 * Class 5
 * Class 6
 * Class 7
 * Class 8
 * Class 9
 * Class 10
 * Class 11 and higher
 * Tetration level
 * Pentation level
 * Hexation level
 * Heptation level
 * Octation level
 * Up-arrow notation level
 * Linear omega level
 * Quadratic omega level
 * Polynomial omega level
 * Exponentiated linear omega level
 * Exponentiated polynomial omega level
 * Double exponentiated polynomial omega level
 * Triple exponentiated polynomial omega level
 * Iterated Cantor normal form level
 * Epsilon level
 * Binary phi level
 * Bachmann's collapsing level
 * Higher computable level
 * Uncomputable numbers

Legion array level

 * Nucleaxul, 200![200200]
 * Big boowa, {3,3,3/2}
 * Great big boowa, {3,3,4/2}
 * Grand boowa, {3,3,big boowa/2}
 * King boowa, {3,3,grand boowa/2}
 * Large boowa, {3,{3,3,3/2}(1)2/2}
 * Super gongulus, {10,10(100)2/2}
 * Super gingulus, {10,100(0,2)2/2}
 * Super gangulus, {10,100(0,3)2/2}
 * Super geengulus, {10,100(0,4)2/2}
 * Super gowngulus, {10,100(0,5)2/2}
 * Super gungulus, {10,100(0,6)2/2}
 * Super bongulus, {10,100(0,0,1)2/2}
 * Super trongulus, {10,100(0,0,0,1)2/2}
 * Super quadrongulus, {10,100(0,0,0,0,1)2/2}
 * Super quintongulus, {10,100(0,0,0,0,0,1)2/2}
 * Super sextongulus, {10,100(0,0,0,0,0,0,1)2/2}
 * Super septongulus, {10,100(0,0,0,0,0,0,0,1)2/2}
 * Super octongulus, {10,100(0,0,0,0,0,0,0,0,1)2/2}
 * Super nonongulus, {10,100(0,0,0,0,0,0,0,0,0,1)2/2}
 * Super decongulus, {10,100(0,0,0,0,0,0,0,0,0,0,1)2/2}
 * Wompogulus, {10,10(10)2/100}
 * Duper gongulus, {10,10(100)2/2/2}
 * Truper gongulus, {10,10(100)2/2/2/2}
 * Quadruper gongulus, {10,10(100)2/2/2/2/2}
 * Quintuper gongulus, {10,10(100)2/2/2/2/2/2}
 * Sextuper gongulus, {10,10(100)2/2/2/2/2/2/2}
 * Septuper gongulus, {10,10(100)2/2/2/2/2/2/2/2}
 * Octuper gongulus, {10,10(100)2/2/2/2/2/2/2/2/2}
 * Nonuper gongulus, {10,10(100)2/2/2/2/2/2/2/2/2/2}
 * Decuper gongulus, {10,10(100)2/2/2/2/2/2/2/2/2/2/2}
 * Hyper gongulus, {10,10(100)2/2/2...100 /'s.../2/2/2}
 * Guapamonga, 10100 && (10100 & 10)
 * Guapamongaplex, 10guapamonga && (10guapamonga & 10)
 * Big hoss, {100,100///...100 /'s...///2}
 * Grand hoss, {100,100///...100 /'s...///100}
 * Great big hoss, {big hoss,big hoss///...big hoss /'s...///2}
 * Bukuwaha, {L,100100}100,100

Higher-order legions
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.
 * BIGG, 200?
 * Goshomity, {100,100\\\...100 \'s...\\\2}
 * Good goshomity, {100,100\\\...goshomity \'s...\\\2}
 * Great goshomity, {100,100\\\\\\...good goshomity \'s...\\\\\\2}
 * Big Bukuwaha, {L2,10}10,10, where X is a Bukuwaha array.
 * Bongo Bukuwaha, {L3,10}10,10, where X is a Big Bukuwaha array.
 * Quabinga Bukuwaha, {L4,10}10,10, where X is a Bongo Bukuwaha array.
 * Meameamealokkapoowa, {L100,10}10,10
 * Meameamealokkabipoowa, {LL100,10}10,10
 * Meameamealokkatripoowa, {LLL100,10}10,10
 * Meameamealokkaquadripoowa, {LLLL100,10}10,10
 * Meameamealokkapoowa oompa, {LLL…A…LLL,10}10,10, A is a meameamealokkapoowa array of L's

Beyond BEAF
These numbers are currently beyond all levels of BEAF ever defined (even those not completely defined or in progress). 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, D5(99)