Climb into Cantor’s Attic, where you will find infinities large and small. We aim to provide a comprehensive resource of information about all notions of mathematical infinity.

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**Sources**

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The number *googol* is the number $10^{100}$, represented in the decimal
system as follows:

10,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000.

The number was named by Milton Sirotta, the nine-year-old nephew of
American mathematician Edward Kasner, and popularized in Kasner’s 1940
book *Mathematics and the imagination*.

A *googol plex* is the number $10^{\text{googol}}$, or $10^{10^{100}}$.
In general, an $x$-plex is $10^x$.

A ‘googol bang’ is the factorial of a googol, that is, the number $(10^{100})!$. In general, an “$x$-bang” means $x!$.

A ‘googol stack’ is the number $10^{10^{10^{.^{.^.}}}}$, where the number of levels in the stack is a googol. This can also be represented $10\uparrow\uparrow\text{googol}$ in the Knuth notation for iterated exponentials.

The plex, bang and stack vocabulary enables one to name some very large
numbers with ease, such as a *googol bang plex stack*, which is the
exponential tower $10^{10^{\cdot^{\cdot^{^{10}}}}}$ of height
$10^{(10^{100})!}$, or a googol stack bang stack bang, and so on.

The *googol plex bang stack hierarchy* is the collection of numbers that
can be named in this scheme, by a term starting with googol and having
finitely many adjectival operands: bang, stack, plex, in any finite
pattern, repetitions allowed.

The initial segment of this hierarchy, using term with at most three adjectival operands, appears below:

- googol stack stack stack
- googol bang stack stack
- googol plex stack stack
- googol stack bang stack
- googol stack plex stack
- googol stack stack bang
- googol stack stack plex
- googol stack stack
- all expressions involving at most one stack, and up to googol-2 many plexes and bangs appear here
- googol bang bang stack
- googol bang plex stack
- googol plex bang stack
- googol plex plex stack
- googol bang stack bang
- googol bang stack plex
- googol bang stack
- googol plex stack bang
- googol plex stack plex
- googol plex stack
- googol stack bang bang
- googol stack bang plex
- googol stack plex bang
- googol stack plex plex
- googol stack bang
- googol stact plex
- googol stack
- all terms involving up to googol-2 many bangs and plexes appear here
- googol bang bang bang
- googol bang bang plex
- googol bang plex bang
- googol bang plex plex
- googol plex bang bang
- googol plex bang plex
- googol plex plex bang
- googol plex plex plex
- googol bang bang
- googol bang plex
- googol plex bang
- googol plex plex
- googol bang
- googol plex
- googol

A sorting algorithm for all such names in the hierarchy involving fewer than googol-2 many terms is provided by this math.stackexchange answer.