Nice effect!

I'll be honest: I got the idea from one of MaRo's columns. He said that it got killed for complexity issues when there were more than one copy out at a time. I figured, why not make it legendary? Then those issues go away.

Very simple solution that makes you wonder why MaRo & WotC didn't come up with it themselves. Well done

So does this thing freak out and empty everyone's hands in a giant loop of discarding, or alternatively, what rules are preventing that?

Pretty sure replacements don't check their replacement for replacements.

Yo dawg! I heard you like replacement effects. So I put a replacement in your replacement effect, so you could replace replacement effects while you replace replacement effects.

What V is saying is "each replacement effect can only apply to the same event once".

So let's say someone casts Probe with kicker.

Unreplaced event: "Draw three cards, then discard two cards. Target player discards two cards."
After Paradox Machine Replacement 1: "Each player draws 3 cards, then you discard two cards. Target player discards two cards."
After Paradox Machine Replacement 2: "Each player draws 3 cards, then each player discards two cards. Each player discards two cards."

Total: Everyone draws 3, then everyone discards 4, just like you'd expect.