The supremum of the cost of the cheese you can purchase is $100.
Perhaps I should clarify. Assume that the ticket has a well-defined value (this may not be the case). Let its value be L.
Is L < $100? No, because you can purchase something worth more than L.
Is L > $100? No, because you cannot purchase anything worth more than $100, including values strictly less than L.
The remaining options are limited.
Yes, but at the same time, we might just as well say:
Is L = $100? No, because you cannot purchase anything worth $100.
So I suppose what I really meant to say was: Whence the principle that its value should be the supremum of the real monetary values it can be exchanged for, even if this is a set with no maximum?
I guess I’m not seeing the issue here. This is a problem of discrete mathematics - there’s no real world value of $99.99999. You jump directly from $99.99 to $100.00 with no intermediate values. So the highest amount less than $100 is $99.99.
I realize the OP said we should treat money as if it had continuous value but that’s not the way money works - it’s always composed of units.
If L is a well-defined quantity, I think it is clear that both L < 100 and L > 100 are not acceptable values. If L = 100 is also unacceptable, you have no choice but to conclude that “the value of the ticket” is not well-defined.
$99.97
Don’t forget about sales tax, guys!
Actually, the conclusion should be that “the value of the ticket” cannot be any real number. As always in math, you have the option of defining a new class of objects that can serve as “values” and then investigating the properties of these new objects. Hence the brief back-and-forth between Indistinguishable and me above.
Regarding the issue of the discreteness of currency: Of course, if you take discreteness into account you’ll conclude that the value is $99.99 (or $99.999 if working to the mill). That’s done; now it’s uninteresting. Hence we move on to the more interesting, if idealized, question of what happens if currency is taken as infinitely divisible.
This is what mathematicians do.
Those coupons that say “cash value 1/100 of one cent”- if the United States does not define any fraction of a dollar smaller than the mill, then how is it valid to claim that something has a cash value less than that?
The ticket is worthless. No one would give me $100 or more for it, because they will then be unable to redeem it for as much as they paid for it. But I would not give it to anyone for less than $100, because however much I give it to them for, I could have gotten more. So, the ticket can never exchange hands (in consideration of its value anyway). Therefore, it’s worthless.
Ignoring the discrete nature of money, there have been three well-reasoned answers proposed:
All of these hinge on particular definitions of “value”; like nearly every paradox, the problem lies in the imprecise use of language. I don’t know what the “right” answer is, but I suspect any progress will have to delve more deeply into the definition of “value” both for this particular problem and in a more general sense.
Worthless like a bag of nothing? This seems an odd position to take. Supposing, out of the kindness of my heart, I were to offer you a choice between either five dollars or this ticket, as a gift; would you prefer the five dollars on the grounds that the alternative is worthless? Are we like some more sophisticated, subtle Buridan’s ass, unable to bring ourselves to ever redeem this ticket for any value, simply because we always could have chosen more?
Holding out forever simply because you can’t get as much as you want for it doesn’t make the ticket worthless; it just means you’re being unreasonably stubborn.
Only if in the hypothetical that one must above all other considerations maximize the redeemed value. The thing about Buridan’s ass to remember is that real donkeys don’t actually do that- they aren’t robotic machines following a logical program even to the point of getting trapped in a loop.
If I’m living in a land of philosophers, I’ll take the five dollars.
But maybe I’m not supposed to assume I live in a land of philosophers…?
You are free to take the five dollars, of course, but doing so while calling the alternative worthless seems odd to me. Why, even in the land of philosophers, would you not take the ticket and then exchange it for, say, ninety dollars? Because you could’ve exchanged it for more instead?
True, you have to make some suboptimal choice.
But note: Taking the five dollars is also an example of making a decision which netted you less than you could’ve gotten; if mere failure to have an optimal choice makes a collection of options worthless, then my gift is just as worthless as the ticket, despite the possibility of presenting five dollars in hard cash.
But why would anyone give me ninety dollars for it?
Well, on second thought, I think we could all get together and plan it out like this: I’ll trade it to you for a dollar, then you can trade it to someone else for two, they can trade it to someone else for three … can trade it to someone else for 99.99999, and they can trade it to someone else for 99.999999, and so on. Assuming an infinite number of people ( 
) everyone profits!
But then, not everyone can profit by the same amount (unless everyone profits by an infinitesimal amount, which I can’t see why anyone would want to do) so there may be no rational way to start the scheme off. The ticket’s back to being worthless after all…
By “exchange”, I meant “redeem”. By stipulation, there is some process by which the ticket is redeemable for ninety dollars (at a bank, say, or the local ticket-mart; or perhaps there’s a button on the ticket you press and it disappears in a puff of smoke, replaced by cash. Whatever it is.).
Oh. Well then, that’s different.
In a situation different from the one given in the OP, though, the ticket’d be worthless. 
(I was thinking of it in terms of exchange because that’s how I think about value by default, but you’re right that this approach is contrary to the spirit of the problem.)
I think I agree that it has no unique real value, especially if we assume the people in this world are immortal and have infinite patience. I put that hedge there because I suspect (but haven’t thought this through) that if people have limits on mortality or patience, then the ticket might have a de facto value constituted by how many times they are willing to undergo repeated transactions a la your post #5.
the supremum is the only number it makes sense to use, if we agree that we must use a number.
this appears to be the constraints on the value α of our ticket:
for all 0 <= α there exists some ε > 0 such that (100 - ε) > α
clearly the set of numbers satisfying these constraints is empty. it remains to decide whether by “value” we require a specific number in this case. that appears to be the OP’s intentions.
in that case, the only value it makes sense to choose is the supremum, 100, for the same reasons we define diameter as the supremum of the all distances among points in an open metric space.