I have seen several questions here on MO regarding large numbers, (uparrow notation, etc.), and different way to construct and compare such numbers.


I am curious what the applications are for the study of such numbers, what is the motivation behind such constructs? There are plenty of constructs that produce large numbers, but the results seems to be of the form *this quantity is finite, but large*, which, surely, is mindboggeling and nice, but it does not reach outside the world of large numbers.


Are there any famous results that use these incredibly large numbers?
In particularly, where the result itself is not just of the form *this number is large*.


I only know some places where such large numbers appear, (Ramsey theory and computability, such as the busy beaver function), but is this motivation enough for study of really large numbers in general? 


Clearly, there cannot be many real-world applications, since these numbers vastly exceed the number of particles in the universe. They are so large that we cannot gasp how large they are, so using them in some application will be fruitless.


EDIT: I would really appreciate examples where the actual number (or big estimate) plays an essential role, not just finiteness of said numbers. 


A hypothetical result I am looking for could be of the following form: 
$f,g$ are special functions defined on positive reals,
such that $f(x)$ has its first zero for some $x$ satisfying $N\_f < x $ and $g(x)$ has its first zero for some $x<N\_g$. Some cool non-trivial result follows only if $g$ is zero before $f$, so this would be implied by proving $N\_g < N\_f$, where $N\_f$ and $N\_G$ are explicit, and very large bounds. 


Thus, I am particularly interested in where working with large numbers is essential for the proof, but the result is *not* of the form "this number is really large".
There are a lot of results of the form *If we do this, then this certain number is really large*, but theorems that involve explicit large numbers where the consequence other than *this is finite, but very large*, seems rare.



Applications of really large numbers
For the conceptual picture that you seem to be looking for, you probably want to think in terms of *fast-growing functions* as the primary object of interest, and very large numbers as evaluations of these functions at specific values of interest. One major reason for the interest in fast-growing functions is that they provide a way to calibrate the logical strength of a system of axioms; the stronger the axioms, the more functions they can prove total.


In Ramsey theory, often the only known way to construct something is via some complicated iteration, and so fast-growing functions arise naturally as measuring the rate of growth of the iteration. Sometimes, one can prove a lower-bound result that shows that the objects in question really do have to grow very quickly, so the large numbers are not just being introduced to give a loose upper bound but are telling you how fast the objects really are growing. One example that Harvey Friedman likes to give is the following.



> 
> **Block Subsequence Theorem**. Let $k\ge 1$. Then there is a longest finite sequence $x\_1, \ldots, x\_n$ from $\{1,\ldots,k\}$ such that for no $i < j \le n/2$ is $x\_i,\ldots,x\_{2i}$ a subsequence of $x\_j,\ldots,x\_{2j}$.
> 
> 
> 


For $k\ge1$, let $n(k)$ denote the length of this longest finite sequence. One can show without much difficulty that $n(1)=3$ and $n(2)=11$. It turns out that $n(3)>A(7198,158386)$, where $A$ is (a version of) the Ackermann function. Now I suppose you could argue that the specific value of $n(3)$ is of limited interest—it's just some big finite number—but the point is that the function used to express it isn't arbitrary; its appearance tells you something about the structure of the combinatorial object in question, and it's also a hint that a moderately strong induction axiom will be needed to prove the result.


