Last I heard, the default limit was twenty-ish, but you can email Support, ahead of time, to get that lifted. I actually see an explicit bound displayed on the dashboard lately:
For thousands of Machines, I’d also suggest consulting the (excellent) new capacity API. Even the largest regions have only ~2000 free.
(This is when the new region-fallback feature comes in handy, I’m guessing.)
Finally, this is just my personal opinion, but at that scale, you’ll probably want a formal Fly Support plan. Most of those include real-time architecture advice sessions, which typically repay the (relatively small) extra outlay on their own—in terms of cost-savings, etc., 
.