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Just when I thought I was out, they pull me back in
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reviewed  Close Colon ideal and ArtinRees lemma 
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reviewed  Leave Open Not too classical group characters 
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reviewed  Close Number of possible wall positions in the game Quoridor 
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reviewed  Close Mathematical definition of running 
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comment 
Tannakian formalism for topological Hopf algebras
This might not be relevant to the categories you care about, but when people try to look at Hopf von Neumann algebras one has to use a different tensor product for the codomain of the comultiplication. So when you speak of Hopf algebras in the category of Banach spaces with its usual SMC structure, which examples do you have in mind? 
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comment 
What is the most useful nonexisting object of your field?
@TimothyChow I suspect I must have noticed your answer there and echoed it in my phrasing :) Actually, I think the phenomenon you refer to is a different idea from the one I had in mind: yours seems to be discreteness; mine is the tension between small and large 
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comment 
Why have mathematicians used differential equations to model nature instead of difference equations
The 2nd sentence in your previous comment does not seem relevant to the 1st sentence, and the 1st sentence still seems debatable. Recall that the point of having a difference equation is to find solutions. How is that simpler than finding a solution of the corresponding differential equation for the continuum limit? Just because the definition is simpler this does not make them easier to work with 
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comment 
What is the most useful nonexisting object of your field?
@Dirk ah, well I tend to use "you can't be both small and big" more than "you can't have two different signs at once" 
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comment 
What is the most useful nonexisting object of your field?
@GeoffRobinson Politicians 
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comment 
Why have mathematicians used differential equations to model nature instead of difference equations
That said, I think there could be good answers to this question; I merely wanted to point out that the starting thesis of your question is open to some debate 
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Why have mathematicians used differential equations to model nature instead of difference equations
Erm, because difference equations are not simpler to actually work with? What is your evidence for saying "they could have been just as successful in modeling natural phenomena with difference equations instead of differential equations" 
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revised 
What is the most useful nonexisting object of your field?
A sensible edit, which I'm just tidying since I prefer to choose my own phrasing (no wiki worship here) 
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awarded  Nice Answer 
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comment 
What is the most useful nonexisting object of your field?
I should perhaps explain that behind my frivolous phrasing is a serious point: see the argument just before the statement of Corollary 4.9 in arxiv.org/abs/0801.3415 , or Lemma 3.6 in arxiv.org/abs/0811.4432 , or Lemma 3.2 in arxiv.org/abs/0906.2253 
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answered  What is the most useful nonexisting object of your field? 
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comment 
What is the most useful nonexisting object of your field?
But if you had formulated your definition using a free group, would you still claim that somehow one is defining a useful nonexisting object? 
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comment 
What is the most useful nonexisting object of your field?
I mean, try replacing ${\mathbb Z}$ in the original question (which hints at invariant means etc) with ${\mathbb F}_2$ 
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What is the most useful nonexisting object of your field?
@NoahS I think it's based on a really debatable premise, myself, and as an analyst who uses proof by contradiction from time to time I can't agree with the first sentence of this question 
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reviewed  Leave Open What is the inverse kernel of this integral transform? 
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reviewed  Close What is the intrinsic geometry of a feasible set? 