Googol

• Googol plex
• Googol bang
• Googol stack
• The googol plex bang stack hierarchy

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.

Googol plex

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

Googol bang

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

Googol stack

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 googol plex bang stack hierarchy

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.