- 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.
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 are believed to have growth rates greater than or comparable to \(\omega_1^\text{CK}\) of the fast-growing hierarchy with respect to a certain reasonable choice of a system of fundamental sequences, but such a system of fundamental sequences is not known.
- \(\Sigma(1919)\) (busy beaver function)
- Fish number 4, \(F_4^{63}(3)\)
- \(\Xi(10^6)\)
- \(\Sigma_\infty(10^9)\)
Rayo numbers
These numbers diagonalize over first order set theory. They are currently the largest named numbers in professional mathematics.
- Rayo's number, \(\text{Rayo} (10^{100})\)
- Fish number 7, \(F_{7}^{63}(10^{100})\)
Large Number Garden Number
This number is currently the largest valid googologism, defined using the first order set theory beyond higher order set theory.
- Large Number Garden Number, \(f^{10}(10 \uparrow^{10} 10)\)
Ill-defined numbers
These "numbers" are ill-defined, i.e. their original descriptions do not define a number.
- BIG FOOT, \(\text{FOOT}^{10}(10^{100})\)
- Little Bigeddon
- Sasquatch / Big Bigeddon
- Hollom's number
- Oblivion
- Utter Oblivion
- Sam's Number
- Infinity
The first three "numbers", BIG FOOT, Little Bigeddon, and Sasquatch, are descendants of Rayo's number, and are considered as significant works compared to the rest five "numbers". As Rayo's function diagonalizes over first-order set theory, the derived FOOT function was intended to diagonalize over nth-order set theory.
BIG FOOT was regarded as a well-defined number, but is actually ill-defined due to several issues. Roughly speaking, there is no "reasonable" choice of axioms which makes BIG FOOT well-defined. See the main article for the issues.
The original definitions of Little Bigeddon and Sasquatch include several obvious errors, and nobody in the community currently understands the way to fix them because the original descriptions lack sufficient information which help us to consider the creator's intention. At least, Little Bigeddon was considered as the largest valid googologism of October 2017. Sasquatch was even guessed to be bigger, but it is also not even known if Sasquatch can be proven to exist; it relies not only on the way to fix the definition, but also on some conjectural statements.
Hollom's number is just introduced as a thought experiment, and hence is not originally intended to be well-defined.
Oblivion is a "number" coined by Jonathan Bowers, but is just "described" in an informal explanation which does not characterise any specific number. If it were well-defined, it would be greater than all the previous numbers. Its extension Utter Oblivion is intended to be even larger.
Sam's number is a "number" coined by a user by the name SammySpore, but isn't defined, only "described". It is obviously not well-defined, but it remains as an in-joke among googologists.
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.