Context: I just completed the first quarter of an Intro to Real Analysis class, and while I was thinking about how some functions (like $x^2$) aren't uniformly continuous because they, roughly speaking, "increase too quickly" (I'm aware of the actual $\varepsilon$-$\delta$ definition), I wondered if there was a "fastest increasing" function as its input goes to infinity (discrete or continuous, it doesn't really matter).
My Research: After a moment of thought, I realized there wasn't one, since we could merely compose it with itself or square it or tack a factorial on the end or do any number of other things. But I was still curious, so I took to the internet and was able to find things like the Ackerman Function and the Fast-growing Hierarchy on Wikipedia. These functions are cool and all, but, as far as I can tell, they don't serve a purpose other than being functions that increase really quickly. So this led to my question...
Question: What is the fastest increasing function that's useful in some area of math? By "useful" I mean something similar to how Graham's Number was used in a proof in a non-arbitrary way. I'm wondering about functions that grow incredibly quickly and exist for reasons other than that they grow incredibly quickly.
Based on my research, it looks some kind of computer science comes into play with these functions, and so do large ordinals. I don't know a lot about these areas because I'm just a second year right now, so don't assume too much background knowledge please.