Measure the 'distance', center-to-center, of the bolt holes of the drain. And just to be safe, measure the INSIDE diameter of the drain too. Take that info w/you to a Plumbing Supply House and ask for what you need. Tell them how much ADDITIONAL height you're putting down on your new floor, INCLUDING the thickness of the tile. From there, you'll get the right part(s).
I could be wrong but something tells me that you can only 'stack' the flanges like 2 at a time. After that, you need an extension vs. a flange. (I'm just putting the 'bug' in your ear. I'm not an expert in this but for some reason, it may be an issue in your situation.) And to boot, I don't know if you have to apply any type of 'adhesive' between multiple flanges/extension, or not. Just mention these 'bugs' and see if they mean anything to the Plumbing Folks.
I've never used multiple wax rings before but I've hear of 2 being used at the same time, not any more than that. I DO know that the key to your 'situation' is to get the flange to the proper height FIRST before you resort to multiple wax rings. Assuming you get the flange to the proper height, you should only need 1 flange (as you have now.) I see it all as being 'relative'. Maybe a Plumbing Pro will stop by here and say more, so stick around for a few days. AT worst, visit the Supply House.