I guess I took this question to be something different. Rather than just "what's the smallest positive value you can think of?" for which I would go with something like:
1-(3↑↑↑↑...↑↑3)
or in other words: 1 over Graham's number (or simply "G")...but why stop there? why not go with
1/G↑↑↑↑↑↑↑...↑↑↑↑↑↑↑↑G
That interpretation of the OP isn't as interesting or fun because it just gets ridiculous... The way I took the original question was pick the smallest positive INTEGER that no one else picks...what is your strategy when up against 9 others doing the same (all allowed to pick however many tokens as they want)?
So, if that's the case, it becomes a much more challenging problem. Obviously it doesn't make sense to purchase more than 9 tokens, as with 10 the BEST you could hope for is breaking even...I would probably purchase the following numbers (and my rationale next to them):
1 - I'm assuming I will lose this $10, but maybe everyone else will think someone else picks this and therefore no one else does!
2 - Might as well try this one as well, but again, most likely will lose this $10 as well...
7...ish - Some relatively low number that hopefully no one else picks...I would assume I'd lose this.
10 - Let's assume everyone buys their 9 tokens and picks 1-9...10 would be the first number that no one would pick...
16...ish - Again, some pretty low number, that's pretty much just a shot in the dark and you hope no one else picks it.
42 - If by some crazy chance everyone buys 9 tokens and all are duplicated, this is the lowest possible number that wouldn't be duplicated...plus isn't it really the answer to everything?
At this point, I've already bet $60 in hopes of winning $100...and really, the odds are still not in my favor...so to me, it's not worth the risk and so my ultimate strategy would be to not play and "break even" :c)