Is there a "largest function"?

In one of my classes, the professor asked about what we think the largest function was. Many thought perhaps ${e^x}^{e^x}$, but I thought about $n!$

When I talk about a "largest function", I mean the function that increases the quickest.

The professor asked about a function larger than $n!$ to which I responded, $2n!$

Although snarky in nature, it is technically true.

So my question is this:

What is the "largest function" if we define "largest" as being "increases the quickest". A parent function is what we need, as it prevents someone like myself from putting a larger coefficient before the function.

Solution 1:

There are many large functions, e.g. $e^n$, $n!$ etc.

And you might know that $e^n$ grows faster than $n^k$ for any $k\geq 1$.

But there are other interesting functions, e.g. the Busy Beaver function. It asymtotically grows faster than any computable function. That means you cannot even write a computer program that produces a faster growing function.

The nice thing is: The busy beaver function is well-defined, but not computable:)! This function really gives an upper bound for the growth of computable functions (e.g., it grows much faster than any function that just contains hyperoperators or the TREE function).

edit: Of course, there are more and even faster growing functions.

Solution 2:

Look into hyperoperators.

https://en.wikipedia.org/wiki/Hyperoperation

This is a sequence of binary operators, each generating larger numbers than the previous. Define $f_n(x) = n \uparrow^n x$. You now have an infinite sequence of functions, each one in the sequence grows faster than the previous one. And they will grow MUCH faster than $n!$.