1. We can think of a greatest conceivable being – call it 'God'.
2. It is greater (qua being) to exist than not.
3. So for any being X to qualify as the greatest conceivable, X must exist.
4. So God [the greatest conceivable being] exists
Distinguish the properties things are conceived to have vs. those they really do have. (Santa is conceived to be a material person, made of flesh and blood, but in fact he's a mere figment of our imaginations.) When we think of the 'greatest conceivable being' in #1, is it actually great? Or merely conceived to be great? If the latter, then the premises are insufficient to establish the conclusion: it may be possible for us to conceive of some X as supremely great (existing) without it actually being so.
In other words: insofar as we can imagine the qualities of a supreme being, one of the things we imagine may indeed be that it exists ("according to the fiction", so to speak). This act of imagination is compatible with the being in question not actually existing -- and hence not actually qualifying as 'supreme'. It's one thing to be imagined as the greatest conceivable, and quite another to really be that way in fact. So, for the ontological argument to be valid, we would have to replace the first premise with:
1*: We can conceive of some being, 'God', and this being is in fact the greatest conceivable being.
But of course if we accept premise 2 then there's no reason at all to think that 1* is really true. There's no reason to think that any being actually possesses the qualities required to qualify as the greatest conceivable (which, after all, include necessary existence).
The ontological argument raises a lot of other interesting issues: whether existence is a predicate, whether it is a perfection, whether there is a unique set of qualities that would render a being insurpassably great, etc. But I think the above counterargument -- based on the distinction between the actual properties of a represented thing vs. the properties it is represented to have -- cuts to the core of the matter.
It's also worth noting that all I've done here is bring out the core insight that finds inchoate expression in the slogan, "You can't define God into existence." I recall there was a bit of a brouhaha a while ago as theists complained about Dawkins' so swiftly "dismissing" the ontological argument on these grounds. But it seems to me that Dawkins was exactly right. It's like Zeno's paradox. The argument is very obviously unsound -- sophistry, even -- and you don't have to be a philosopher to see this. The difficulty simply lies in pinpointing the error, and explaining it in a philosophically sophisticated way. Any old fool can recognize that there's an error there to be found (they even have a rough sense of what the error is).
Granted, the ontological argument is philosophically very rich, and rewards further discussion and exploration in the philosophy classroom. But that's not because there's any real question as to its soundness (as would be relevant to a popular book on the question whether God exists). It's simply because the mistakes - and the further questions it raises - are so subtle and interesting. It is, in short, of purely academic interest. (Not that there's anything wrong with that.)