An atheist (here) says that the ontological argument fails because, "what it hinges on is the definition of God as being "maximally great" and of maximally great as including the idea of necessary existence. Necessary existence simply isn't logically possible." When asked for a reason to think this, he said, "Something cannot be logically necessary if it's contrary is logically possible. It certainly appears that a world with no God is logically possible." But this is demonstrably false.
When considering the ontological argument, you have to differentiate between metaphysical possibility (actualizability) and epistemic possibility (imaginability)1. The statement that it appears that it is logically possible for there to be a world with no God only has epistemic merit, in that we could think of God not existing. To say, however, that it is possible for a necessary being to not exist is truly square circle territory metaphyiscally. How could a being that is necessary not exist? While it may be possible to imagine it, it is impossible to actualize a world where a being that must exist doesn't. That is why the ontological argument works. It shows that God is either necessary, or impossible. So, David must show that it is impossible that a maximally great being exists. But it certainly seems possible...
Furthermore, the statement, "it is possible that God does not exist," is not the negation of, "it is possible that God exists." Rather, the negation of "it is possible that God exists" would be "it is not possible that God exists." So his contention fails at that point as well.
1 William Lane Craig and J.P. Moreland, Philosophical Foundations for a Christian Worldview (Downers Grove, IL: IVP Academic, 2003), P. 497