I read once that every number is interesting. However, I was disappointed when I read things like "$17$ is unique because it's the smallest integer one larger than an even positive fourth power" -- that doesn't make $17$ unique, I thought, since "one larger", "even", and "fourth power" are arbitrary choices; you could replace them with any difference, any divisibility, and any power respectively and get another "interesting" number. Unique? Hardly.
However, when I moved away from arithmetic into other fields, I began to see some really interesting numbers, or at least numbers appearing in interesting and unique places; for example, the line graph of the complete graph $K_n$ is determined by its spectrum, except in the case of $n = 8$. As another example, there are exactly $27$ (I'm counting the Tits group as sporadic since it is not technically of Lie type) simple groups that are not cyclic, alternating, or of Lie type. With all the strange literature behind these two facts alone, you cannot deny these two numbers as interesting, and wholly unique in their own respects.
Upon further pondering, I realized that the facts I was looking for were places where numbers appeared in ways that they could be defined by without having to define any other numbers. We could define $8$ as the eighth successor of $0$, sure (never mind the circularity), but we could also define it as the only $n$ where $L(K_n)$ is not determined by its spectrum, and without having to even define $0$ or any other numbers (although we would have to define a bulkload of graph theory and linear algebra). You can define $3$ as the first $n$ such that a bug taking random unit steps could get lost in $n$-dimensional space. No arithmetic needed: only abstract mathematical concepts (yes, arithmetic is also an abstract mathematical concept, but I'm looking for non-numerical places where numbers show up uniquely).
I'm simply looking for more examples like these: places where a number shows up in a unique way, and where it's not there by arbitrary choices of parameters or simply mashing together other numbers, but appears in a fundamental way. It could be a counterexample to something, but it doesn't have to be -- for example, the number of sporadic groups is not a counterexample but a unique quantity of fundamental objects.
(I'm aware that "arbitrary" and "fundamental" are arbitrary concepts themselves, but use your intuition here. Does the idea of taking the fourth power specifically seem arbitrary to you? To me, it does. How about simple groups? Less so. Maybe you have different opinions about the semantics of arbitrariness but if you think it would make an interesting answer feel free to post it.)
Let me know if this question is badly written or confusing or off-topic in any way (and feel free to edit the tags as you see fit, I didn't really know how to tag this). I look forward to seeing what interesting examples this community can give!
