Two of the oldest arguments against the existence of God, in the sense of perfect being theology specifically, are the problem of evil and the paradox of omnipotence. The former argues that if there were a God, there would be no evil in the world, but there is evil, therefore there is no God. The latter goes after the concept of God, arguing the idea of being all-powerful makes no sense. A standard version of this second question is "could God make a rock so big even he couldn't lift it?" but a clearer version comes from J. L. Mackie's article "Evil and Omnipotence" (which deals with both problems): can God place limits on himself?
A number of atheists, myself included, have found the problem of evil significantly more compelling, at least on an intuitive level. Why? One temptation is to say that the problem of evil invokes evidence from the observable world, but this is a flimsy rationale: standard versions of the problem of evil involve at least some deduction from abstract principles, and if an idea is incoherent, one shouldn't be stopped from saying so by fears of being accused of armchair reasoning. Another was suggested by Richard Carrier in his book Sense and Goodness Without God: criticisms of the coherence of theism just end up being arguments for the reform of theology. This doesn't seem to hold much weight either, though: the problem of evil can also be evaded by changing your theology.
Another answer to this quandry was suggested to me the other day when I was reading Peter van Inwagen's Giffod Lectures, which deal with the problem of evil. The book on the whole isn't worth buying, as the main material is available elsewhere. However, it does have an interesting section on the divine attributes. Van Inwagen scales them back a bit. For one, he rejects the idea that God knows what free agents will do in the future, a move considered wildly heretical in some circles but which is nonetheless gaining in popularity. More surprising, he rejects the idea that God can do anything logically possible (a common purported response to the paradox of omnipotence), on the grounds that the concept of logical possibility is problematic. Then he tries to draw a distinction between accaptable and unacceptable redefinitions of God. He claims that an understanding of God's attributes must be "loyal to the idea of God as the greatest possible being." This, I think, pretty well captures what theists want, so even if you can prove one concept of God incoherent, they can just fall back to the greatest concept they can get away with.
Maybe, though, you can go after van Inwagen's minimal concept. Maybe you think there are two different but incompatible ways of being extraordinarily powerful (Mackie suggests something like this), or maybe you think for any being there is, there can always be a more powerful one. What are theists to do here? I think many, though they would not say it openly, would be willing to settle for one of the greatest beings they can imagine. That's close enough, right?
Now we see the power of the problem of evil: not only does it show there is no greatest possible being, it shows us how to imagine a being greater than any one that actually exists, by imagining a being that prevents at least some of the horrendous evils that have, in unfortunate fact, been visited upon the world.