Is God supernatural?

A redated post. 

I wouldn't even necessarily call God supernatural. There could conceivably be a science studying God's actions, based on which we could make predictions. If God would let us, we could even perform experiments on Him. What's wrong with this idea?

Actually, if you say that what we mean by supernatural is that it won't fit in to a mechanistic order, then of course God is supernatural. That's how Lewis defined it. But does everyone understand the term "supernatural" in that way? It seems like a lot of baggage is brought into the use of this term.