No problem. I've added an edit to my answer to give something reasonably simple (I guess).

I will defend my answer as giving a class of examples, and also of putting the idea of weak initial set into a motivated context. If you want the simplest example, then perhaps you should say so?

Sorry, David, I'm not sure what you're worried about. Any field contains a smallest field (the smallest field containing 1) where 1 is either torsion (making the smallest field a finite field F_p) or not (making it Q).

@arsmath: yes, when the random mathematician is asked to describe cocompleteness, he already has in mind some external notion of set in the background; cf. my comment to David Roberts's answer. (This isn't arguing with you; I happen to be interested in an honest discussion of these issues.)

Why can't I get a URL here?

Sorry: ncatlab.org/nlab/show/cocomplete+well-pointed+topos

It may be relevant to remark though that an elementary topos is also "cocomplete" relative to definable families internal to the topos; see for example the discussion on the nLab here: ncatlab.org/nlab/show/cocomplete+well-pointed+topos. So the cocompleteness we are discussing here is external cocompleteness, which depends on a background notion of set.

@arsmath: yes, your union of iterated power sets is the classic example of a set that cannot be formed without replacement. In fact, starting with a universe $V$, the subuniverse of sets of ordinal rank less than $\omega + \omega$ is a model of ZC (remove the F = axiom of replacement). But I don't see how this construction is particularly needed by mathematicians who are not set-theorists. I would rather know: are there important constructions used in core mathematics, not counting the needs of set-theorists, which really require replacement?

@Andras: Joking aside, there is a serious point behind Taylor's post: that the replacement axiom is indeed enormously powerful, even if not strictly speaking necessary for most of core mathematics (which can be developed within a universe of sets whose rank is less than $\omega+\omega$, where replacement fails). In fact, Taylor's book Practical Foundations of Mathematics culminates in a discussion of how to formulate the replacement axiom in the context of dependent type theory.

@Harrison: interesting question(s). My first wonder is whether it's even (intuitively) true, that most theories are inconsistent; I can't make up my mind. One could ask similar but perhaps simpler questions like, "are most group presentations presentations of the trivial group?" Here I think my instinct leans more towards saying that "most" group presentations, if not of the trivial group, are undecidable.

(I took Doyle's depiction of Conway's objection to the form of the paper -- all the "fluff" -- as referring more to the very discursive expository style.)

I would fully agree, except that I've heard Conway say similar things in person.

Um, yes, I know all this, Andras?! Was joking around myself?!

Kevin Buzzard: yup. See mta.ca/~cat-dist/catlist/1999/zf-010499.

Compare Pierre Cartier, as quoted by David Ruelle in Chance and Chaos: "The axioms of set theory are inconsistent, but the proof of inconsistency is too long for our physical universe."

What opinion? That ZFC or even Peano Arithmetic is in fact inconsistent? The trouble is that this sort of thing is not too likely to be put in print by a reputable mathematician, even if he expresses his (perhaps occasional) doubts in private. But yeah, I can think of a few reputable mathematicians who sometimes express sentiments ("opinion" is maybe too strong a word) along these lines. Backing that up is another story altogether.