@ Dmitriy, we had a similar issue with a store, but clearing the cache solved the issue. In Elizabeth's case, it didn't work. Ours was when you would be in the download page, for example, and refreshed the page, a new order would appear. And it only happened in Chrome! It's gets even more strange. I would generate order numbers that would skip 4 or 5 numbers ... like 78, then 82 then 87 etc.
What it was doing was giving that individual free downloads of the same product!
Just and FYI... there seems to be some issue with the download products.