- Class 0 and 1
- Class 2
- Class 3
- Class 4
- Class 5
- Tetration 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
The term "uncomputable number" here refers to the numbers defined in terms of uncomputably fast-growing functions.
Note: The special cases of the iota function and Hollom's number are not listed due to ill-definedness.
Busy beaver numbers
These numbers arise from functions that eventually dominate all computable functions, and are based on the unsolvability of the halting problem. They exploit the maximum scores of a particular Turing machine, or related systems, given the condition that they will halt. They have growth rates of at least \(\omega_1^\text{CK}\) of the fast-growing hierarchy.
- \(\Sigma(1919)\)
- \(\Xi(10^6)\)
- Fish number 4, \(F_4^{63}(3)\)
Rayo numbers
These numbers diagonalize over nth-order mathematical theories: Rayo's function diagonalizes over first-order set theory, and the derived FOOT function diagonalizes over nth-order set theory. They are currently the largest named numbers in professional mathematics.
- Rayo's number, \(\text{Rayo} (10^{100})\)
- Fish number 7, \(F_{7}^{63}(10^{100})\)
- BIG FOOT, \(\text{FOOT}^{10}(10^{100})\)
- Little Bigeddon
- Sasquatch / Big Bigeddon
BIG FOOT was regarded as a well-defined number, but is actually ill-defined. Little Bigeddon is considered the largest valid googologism as of October 2017. Sasquatch is even bigger but the community currently cannot understand it. However, it is also not even known if Sasquatch can be proven to exist; it relies on some conjectural (although likely true) statements.
Oblivion
Jonathan Bowers defined a number called "Oblivion", but the well-definedness is debatable, but if it was well-defined, it would be greater than all the previous numbers. Even larger is "Utter Oblivion".
Sam's Number
A user by the name SammySpore created a page called "Sam's Number", but the "number" described isn't defined, only "described". It is obviously not well-defined, but it remains as an in-joke among googologists.
Infinity
Infinity is not a number. It is not considered a googologism of any sort, and googologists don't like people messing with it in googology. However, transfinite ordinals (a set-theoretic type of "infinity"), are sometimes used to index functions.