Hey Jeff,
Yes, I always have water in the bog. I let the level go up and down, but don't let it drain completely. My bogs rest in larger bowls of water, so every couple of weeks I dump the water out and replace it with fresh water.
And yes, the cost of distilled water becomes a factor after a while, especially if you have several mini bogs as I do. I have two ways of controlling the costs, first, I buy "reverse osmosis" filtered water from a vending machine outside my local supermarket for 35 cents a gallon, and second, I've purchased a reverse osmosis filter for my home, which I haven't installed yet. With just one mini-bog, however, you don't really go through that much water, unless you're in really hot, dry conditions.
Capslock