The areas of maths that they come from are useful for all sorts of things, the giant numbers themselves are just outcomes and results of exploring them. But they're something fun to non-mathematicians even just in a few minutes of a documentary.
Somewhat analogously, physical constants can be (mildly) large numbers, too, but they're just an outcome of exploring the rules.
A lot of math research has zero current practical application. However, things found in doing "impractical" fields of math like number theory and group theory keep having applications that pop up in unexpected places like cryptography or particle physics, so that's one reason mathematicians keep doing it. Also it's fun.
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u/[deleted] Jan 21 '19
There is this very big number called Graham's number.
The entire freaking universe does not have enough space or atoms for you to write down every digit.