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I think that it is only multiplied by x2, no matter how many x2 multipliers you buy.
I am assuming that if you buy one full set of x2 hats, you get the silver sling. You’re next full set gets you a gold sling. Your third full set gets you whatever comes next. (But because a full set of cowboy hats apparently gets you an azure sling, I have no idea what comes next.)
As I’ve shown, under the current conditions of the game, you only face opponents with the same sling. So I don’t see the point at all to add more modifiers (hats) of the same level. If you already have one full set of x2, then move on and start buying x3.
Now that we can see which hats the other players are using, it’s even more clear to me that the strange hat usage we see from our opponents (e.g., silver sling but many birds with party/no hats) is not real. It’s most likely manipulated post-run, server-side, by rovio.