Numbers like $(10)^{-10^{10^{10}}}$ are generally regarded as small, whereas numbers like, for example, Graham's Number, are regarded as extremely large. My question is, are these numbers simply "small" and "large" relative to "every day" numbers or are the notions of "smallness" and "largeness" absolute in some sense, meaning that "small" and "large" numbers can be objectively categorized in a meaningful mathematical way.
If the answer to this is, as I am presuming, no, that it is simply relative, then is it also true that the words "small" and "large" have no relevance in pure mathematics, where "every day" numbers are generally not privileged? What significance exactly do the symbols $>>$ and $<<$ have in pure mathematics? Or sayings like "for small $n$" or "for large $n$"?
The symbols $\gg$ and $\ll$ don't have a formal definition. Usually they are used to compare two extremely big numbers, for example $\mathrm{Graham's \; number} \ll \mathrm{TREE}(3)$ or something like that. They are only used because someone wants to make clear that one of them is so much greater.
The numbers are indeed just small compared to everyday standards. Because when you compare $10^{-10^{10^{10}}}$ and $10^{-\mathrm{Graham's \; number}}$, you can see that $10^{-10^{10^{10}}}$ is actually much larger.
The saying [$P(n)$ for small $n$], where $P(n)$ is a proposition, formally means that there exists a $\varepsilon > 0$ such that for all $0<n<\varepsilon$ we have that $P(n)$ is true.
Similiar, the saying [$P(n)$ for large $n$], where $P(n)$ is a proposition, formally means that there exists a $M > 0$ such that for all $n>M$ we have that $P(n)$ is true.