Ok, so god would need to be, at a minimum, something that is not confined by laws of nature and the empirical world, i.e. supernatural.

I think I can perfectly well say "god does not exist," because I'm still using the definition of existence we apply to any other subject of that predicate. The statement can be rephrased as "god cannot exist," because, as you yourself agreed, god does not exist within the boundaries of the natural world.

"I would also say we can't say something doesn't exist just because we are unable to investigate it scientifically."

But would you say "we can't say something doesn't exist just because it is absolutely impossible, ever, for us to investigate it"? You yourself said that god cannot offer any evidence of himself that we would not be able to interpret as a natural phenomenon, you know, card tricks. Is this not the most egregious violation of Occam's Razor possible, to postulate an entity that nobody can ever experience as a causal agent? Such an entity offers no explanatory power, we can just as easily postulate the phenomenon itself as fundamental. What if I told you that rain is some perfectly elusive sky-clown pissing all over the world, would you say "yes, it's possible"? And even if you would say that, what difference would this make upon the world or our lives? I would claim none.

"Another point, because I think I let our definitions drift a little -- I wouldn't say that we can't define existence outside our universe, but rather that we can't describe it. So you could counter this if you could show that "existence outside our universe" is meaningless in the same way that "temperature below absolute zero" is meaningless."

Ok:

"Temperature below absolute zero" is, by definition of the terms, a meaningless statement. Temperature is a metric, and the absolute zero is the bound of that metric. Applying the word "temperature" to describe a state outside the bounds of its definition is nonsensical, therefore "temperature below absolute zero" is meaningless.

"Existence outside our universe" is, by definition of the terms, a meaningless statement. Existence is a property, and space-time is the bound of that property. Applying the word "existence" to describe a state outside the bounds of its definition is nonsensical, therefore "existence outside our universe*" is meaningless.

*should read "existence outside space-time", perhaps there are universes other than our own, but existence in those universes would still be bound by space-time.

I think you're misunderstanding my position a little when you say, "The statement can be rephrased as "god cannot exist," because, as you yourself agreed, god does not exist within the boundaries of the natural world." I didn't say that a god would not exist within the boundaries of the natural world; I said that a god would not be confined by the laws of nature and the empirical world. This is a big difference, and one we keep coming back to. I feel that restricting supernatural things from interacting with empirical things by definition is arbitrary. Saying that we cannot venture into a supernatural realm does not imply that a supernatural realm could not interact with us, even if we would perceive those interactions empirically. Similarly, I would argue that our inability to interact with a fifth dimension of space would not prove that there is no such thing.

You ask if I would say that something doesn't exist just because it is absolutely impossible, ever, for us to investigate it. No, I wouldn't. It might be impossible for us to detect alternate universes, but I wouldn't conclude from this that they don't exist. If something came before the big bang (so far as that term is meaningful), it is impossible to investigate, but I wouldn't say that the big bag was preceded by nothing. I wouldn't even make the statement that the big bang was necessarily causeless (keeping in mind that there could be a natural cause, of course).

Is it a giant violation of Occam's Razor to postulate something nobody can ever experience as a casual agent? Well, I don't agree with the implication that a deity wouldn't be able to cause experiences, but in any case, you're right that a deity is, to me, a giant violation of Occam's Razor. But violating Occam's Razor doesn't prove that something doesn't exist, it just makes it the less favorable explanation (in this case, the much, much, much less favorable explanation). Also, someone with different knowledge or assumptions about probability might reach a different conclusion (as is often the case).

As for the sky clown, how are you defining it? I'm assuming that we can't go and check for the clown, so perhaps you are using the term "sky clown piss" in the same way that a Catholic uses the phrase "body of Christ" during mass. If so, then I can't prove you wrong, but we would have to spend some time defining your terms before I was sure you were even saying something meaningful.

What difference would the sky clown make to our lives? You're right -- none (at least for practical purposes). But that doesn't prove anything. Of course, if we're talking about God, whether or not he exists makes a lot of difference to a lot of people. In fact, if a deity of some sort exists it might make a big difference to you and me, too, whether we know it or not.

I see what you mean with the proof that existence outside space-time as meaningless, but I don't necessarily agree with your definition of existence. Also, allowing for other universes to be included within this definition because they have space-time (even though it might not be compatible with our space-time) seems to make the issue even more complex.

When I spoke of a deity not being bound by the laws of nature and the empirical world, I was talking about our universe's laws of nature and the world we can detect. I would not rule out the possibility that a deity exists within some kind of universe. For example, a higher-dimensional being that created and can interact with our universe would fit my definition of a deity, even though it would have an empirical world of its own. I could not describe the properties of such a being, and I can't imagine what it would be like, but it seems to me that such a concept is not logically impossible.

Now, if you would consider this higher-dimensional being not to be god because it has a space-time existence in a context of its own, and you cannot conceive of anything without a space-time existence being possible, then I think you might be justified in saying you believe there is no god. But you would be doing so with a definition of god that might not be universally acceptable.