Theists: What is the most compelling argument you have heard for Atheism?

[SteveII]

The multiverse doesn't have to be finely tuned because it can be thought of as a kind of "foam" in which different "bubble" universes come to exist with their own physical constants. As aforementioned the multiverse was hypothesised based on our current understanding of physics, so it is at least somewhat viable in comparison to the idea that an intelligent designer did it. [1]

As for the claim that if the physical constants were slightly different then our universe couldn't evolve to have structure, I agree with that completely, but it doesn't explain anything because it only applies to OUR universe. The fact that the physical constants are the way they are is a consequence of the fundamental nature of our universe, not the other way round, so by definition they can't be different. Constants don't give rise to universes, they are a property of universes. The nature of other universes could be so vastly different that they may have different constants, or they might not even have constants. The correct question to be asking is "how likely is it for a universe with structure and significant duration to come to exist" and sadly we just don't know the answer to this question.  [2]

The fine tuning problem also cannot explain why the universe appears to be overly tuned in some aspects. For example, the entropy at the instant of the Big Bang is many orders of magnitude lower than what it needs to be for the Stelliferous Era of the universe to last up until now. I would appreciate if any theist here could explain why God would need to fine-tune the Universe to an unnecessary extent. [3]

1. The mechanism that generates universes must itself has laws that govern how universes are spawned. An inflationary-type multiverse must have the following mechanisms: 
i. cause the expansion of a small region of space into a very large one.
ii. generate the very large amounts of mass-energy needed for that region to contain matter instead of merely empty space.
iii. convert the mass-energy of the inflated space to the sort of mass-energy we find in our universe
iv. cause sufficient variations among the constants of physics to explain their fine-tuning 

Both (i) and (ii) are achieved by two factors: a) a postulated inflation field that gives empty space a positive energy density, and b) Einstein's equations from General Relativity. (iii) requires the old E=MC^2. So before we even get to creating matter/laws/constants, we have a very precise initial conditions necessary to to create a "random" universe. How is that not fine-tuned?  

 (reference: Blackwell's Companion to Natural Theology, p.263 ff)

2. I don't disagree with your distinction. It is your last sentence in that paragraph that is the question: how likely are we to see these constants? Epistemic probability says that is is unreasonable to postulate mere chance and therefore supports the idea that the constants were designed to be the way they were. 

3. Actually, the low entropy of the initial conditions is a problem for inflationary cosmology (which in turn is the reason for postulating a multiverse). They have to compensate in their models for it (and don't know why).