My dad has more money than yours.

In this well known children’s game, both player name an integer, and the winner is the one who names the largest integer, or better still the one who utterly humiliates his opponent by naming a vastly larger integer. Integers must be named in a constructive way, say by writing down an explicit Turing machine which halts after a large number of steps (so for example values of the Busy Beaver function are not allowed, as this function cannot be defined constructively).

The problem is to find a good strategy for this game, in other words to give a (constructive) definition of a very large integer.

Any easy way to write down a large integer is to write down a rapidly growing function, and take some value of this function.

If the only functions we use are addition of 1, (or addition of some fixed constant), we get numbers like 1+1+1+1+1+1 or MMMCCXXXVIII.

If we allow addition and multiplication as operations we get numbers such as 18362528394746. This covers essentially all numbers needed in physics (unless one wants to know how long it will take for a large black hole to decay by Hawking radiation, or something like that). A common mistake made at this level is to write a long sequence of nines: since ones are thinner it is better to write a long sequence of ones.

The next step is to allow exponentiation. This allows one to write down numbers such as Skewes number

This covers almost all numbers used in mathematics.

Exponentiation is repeated multiplication, so it does not take much imagination to define an operation of repeated exponentiation, and so on. Repeating this idea gives the class of primitive recursive functions. These are roughly the functions that can be defined by starting with the successor function and allowing functions to be defined by recursion over the integers. A typical example of a number produced with such functions is Graham’s number, which seems to be the largest number that has turned up “naturally” in mathematics (rather than by someone deliberately trying to produce a large number).

The next step is functions such as Ackermann’s function, one of the simplest functions that is not primitive recursive. Primitive functions allow one to define functions by recursion over the first countable ordinal ω, but Ackermann’s function allows one to use recursion over the ordinal ω×ω=ω². To define bigger functions, one can use larger (countable, constructive) ordinals. Some obvious ways of constructing these are ordinal addition, multiplication, and exponentiation.

The smallest ordinal one cannot form in this way is called ε₀, the limit of

which is the ordinal measuring the strength of Peano arithmetic, in the sense that it is the smallest ordinal that Peano arithmetic cannot prove to be well ordered (Gentzen’s theorem). A reasonably natural function at this level, related to Ramsey’s theorem, was found by Paris and Harrington as part of the proof of the Paris-Harrington theorem .

More generally, for any theory extending Peano arithmetic there is a countable ordinal measuring its strength, defined as the smallest (constructive) ordinal that the theory cannot prove is well ordered. There are also some associated rapidly increasing functions, which one can in fact define directly from the theory without mentioning ordinals: take all computable functions f_n that the theory can prove are well defined, and “diagonalize” over them. (More precisely, define a new function f(n) = 1+max_i,j≤n f_i(j); this new function is increasing and eventually larger than any of the functions f_n.)
This produces a function that is well defined (at least if the theory is ω-consistent) but that grows so fast that the theory cannot prove it is well defined.

Incidentally, this gives a proof of one version of Godel’s incompleteness theorem: the theory cannot prove that the function f is well defined, as it is larger than all functions it can prove are well defined, but the ω-consistency of the theory easily implies f is well defined, so the theory cannot prove its own ω-consistency.

So to find large functions we need strong theories. Beyond Peano arithmetic there is second order arithmetic, followed by various set theories, such as Zermelo-Fraenkel set theory. We can go further using the hierarchy of known large cardinal axioms, the largest of which is the axiom for Reinhardt cardinals. (These are so large that they are not consistent with the axiom of choice, so one has to use ZF (Zermelo Fraenkel set theory without the axiom of choice) rather than ZFC.)

So to summarize, we get a really large integer as follows: take Zermelo-Fraenkel set theory with a Reinhardt cardinal. List all computable functions that it can prove are well defined, and diagonalize over them. The value of this function at some reasonably large integer (say a trillion) will be larger than any (constructive) number written down by anyone without training in logic and set theory. It is necessary to take the value at quite a large integer because the function starts off growing quite slowly, since most functions that the theory proves are well defined are quite small.

There is one problem: it is not clear that this function is well defined, and there is no known way of proving that it is. This is inevitable: any function that one can see is well defined must be quite small.

(Added later): if one allows numbers that are just defined, rather than constructively defined,
one can do much better; for example, the busy beaver function at reasonably large values will exceed all the numbers mentioned above. It is easy to define very large numbers from powerful theories with a minor variation of the ideas above as follows: for each theory one can define a function f(n) to be the largest number that the theory can specify and prove is well defined in less than n symbols. For example, one could take the largest number that Zermelo-Frenkel theory with a Reinhardt cardinal can define and prove is well defined in at most a googolplex symbols.