Clearly, any number can be trivially given compact notation in some system. Just say “Alright, that number? I’m going to represent it as ‘R’ from now on. There; just one letter to express it, in the particular notation system I just made up”.
It’s only when you start asking more of the notation system than simply “Has a short name for this particular number” that there’s any kind of non-trivial design problem. If you don’t impose any further requirements/constraints, then it’s easy.
Of course, but I obviously can’t give you an example. Consider, though: There is a finite amount of matter in the observable Universe, which can be arranged in a finite number of ways. Even if I could consider every possible arrangement of matter in the observable Universe to be a representation of a number, that’s still a finite list of numbers that can be represented. Any finite list of numbers must contain some largest element, so any number larger than that would meet your criterion.
And it’s no fair saying that you could just use some different notation to make the same arrangement of the Universe represent a different number, since you need to specify your notation somehow, and whatever you’re using to specify the notation must itself be part of the observable Universe.
Well, much like in the thread linked below, let me just note there is a difference between “For every notation system, there is some number which lacks a short name within it” and “There is some number which lacks a short name within any notation system”. The former is obviously true and the latter is obviously false.
But to say “You need to specify your notation system somehow” is to require an infinite regress… in what system should I specify my notation system? And in what system should I specify that system? And so on. (I spoke a bit about the “Wittgensteinian” resolution of this issue here)
Which is to say, if you really stick to your guns with the principle “You need to specify your notation system somehow” (and don’t just off the bat pick some particular notation system as privileged and not in need of specification), then all descriptions become infinite: one must augment the notation N_0 of a number with some notation N_1 of the notation system in which N_0 is to be interpreted, which must itself be augmented with the notation N_2 of the notation system in which N_1 is to be interpreted, ad infinitum.
(And, of course, in the end this still doesn’t clear up ambiguity over exactly how all these notations are to be interpreted, as two people could still go on interpreting every link in the chain differently. Which just goes to show that this kind of requirement accomplishes nothing and isn’t the right way to think about what counts as notation.)
(If one does pick some particular notation system as privileged, then, of course, there are numbers which cannot be given short names within it, no matter what, even with such techniques as auxiliary definitions of new languages or what have you given in the privileged notation system. But this is just “In every particular notation system, there are numbers which lack short names” again. It’s not as though those numbers are intrinsically unwieldy to express in themselves; they’re just unwieldy to express relative to that notation system. To the extent that privileging that system over others is arbitrary, so is the predicate of being easily expressed within it as opposed to within others.)
The number of thy answer shall be three.
No more, no less.
Four would be right out.
As would be 2.
That your “conjecture” is the first, but not the second. Nobody (and certainly not the mathematical community) thought that this proposition might be true and then settled the question by discovering a large counterexample. The falsehood of the proposition follows from Peano’s axioms in just a few steps. In fact, if “conjecture” is to mean something more specific than “proposition” (as it is intended to), your “conjecture” is not, and was never, an conjecture at all.
I’ll grant you that no one ever thought the conjecture was true [well, I recall once as a very young child confidently stating something along such lines to my sister (though with an embarrassingly far more modest upper bound)…]. But, I say, the only reason everyone knows it to be false is because they all readily see how to discover a very large counterexample. So as far as I’m concerned, it’s both the first and the second. 
(And if we settle Goldbach’s conjecture by finding a counterexample, it will be both the first and the second as well)
Or even entertained the notion that it might be true. So it’s not a conjecture, and is disqualified on that ground alone.
This is an attempt to rob a legitimate question of its meaning.
And I can dismiss all claims of the form “all numbers are less than or equal to n”, without finding a counterexample of any of them.
Sure, and that might become the answer to the OP. But your pseudo-conjecture is not a candidate.
(Graham’s Number) + 1
I don’t really care about whatever argument we seem to have going. However, out of curiosity:
I’m sure you can, but I imagine the first idea that occurs to most people is “Well, there’s always n+1” (see above), which certainly is finding a counterexample. [And, of course, there are myriad ways to reword this sort of thing]. Still, what would your approach be?
OK, good. I hate those pointless acrimonious arguments too.
I may not have been clear. I didn’t mean that I could dismiss any such proposition. I meant that I can, right now, simultaneously dismiss every such proposition. The reasoning is obvious enough. It could be done by noting that every number has a successor, and for every number’s successor is bigger than that number. And that’s really all you’re doing in your example, but in a narrower way.
Consider the following:
Proposition: There is no number larger than 1,000,000,000!.
“Counterexample”: (1,000,000,000!) + 1
Proposition: There is no number larger than the billionth prime
“Counterexample”: The billionth prime plus one
Proposition: There is no number larger than the first number for which Goldbach’s conjecture fails, if there is one
“Counterexample”: (The first number for which Goldbach’s conjecture fails) + 1
In none of these cases have we found a counterexample in the sense of the OP.
Or, indeed, (Graham’s Number)[sup](Graham’s Number)[/sup].
Raised a Graham’s Number of times. :eek:
Graham’s number -> Graham’s number -> Graham’s number.
Those who are interested in discussing large numbers should check out my old thread On large numbers.
Or, to steal an idea from xkcd:
calling the Ackermann function with Graham’s number.
A(g64,g64)
example of how quickly the Ackermann function grows (from the Wikipedia page)
A(4,3) = (approximately) 10^(10^19727.78)
Oh yeah? (Graham’s number -> Graham’s number -> Graham’s number) -> (Graham’s number -> Graham’s number -> Graham’s number) -> (Graham’s number -> Graham’s number -> Graham’s number)
So there 
Bah! The largest number specifiable in [your fixed language of choice] in less than [your upper bound of choice] characters. Boy, it’d take you [longer than your upper bound of choice] to beat that!
D’oh!