I enjoy the debate, and fully admit I may be wrong. An infinite number of monkeys will create Shakespeare, but cannot produce a list of all typewritten lists that do not contain themselves as it is simply a paradox, not an improbability.

This is why I compare to pi. You can find an infinite number of 1s, infinite number of 14159s an infinite number of a million digits of pi, but it does not repeat. So if you say an exact earth is 15 digits long then yes you will find an infinite number of those digits (and every other combination of 15 digits). But you will not find anywhere that pi starts at 14159 runs any distance and then repeats 14159 and on exactly. How does this relate to the copy of earth? Earth has influenced and been influenced by everything within 14 billion light years, so 14 digits of pi let's say, you can certainly find an infinite number of those, but at the edge of that 14 billion years those influenced objects have been influencing things for 14 billion years as well. (I'll grant that we're probably getting into less than plank lengths so I'll admit I'm wrong here, but hear me out anyway) the continuous expansion of the sphere of influence encompasses the entire infinite universe in this way. So in order to create an exact copy, with the same history, same future, same influence you need a copy of the whole universe, the entire length of pi... And pi does not repeat, though it is infinite.

Anything less than an entire universe copy will eventually diverge from our earth (perhaps after the heat death of the universe and be really really really hard to notice but would be different none the less)

If you are talking about a multiverse, infinite big bangs, then clearly yes this happening once is proof it is possible, and would happen infinite more times.

If the size of your copy is infinite you need a new infinite universe to put it in.

The other point I'd like to raise is that not everything is random, so something like the same earth but I'm left handed may not be a possibility. The genetics of my parents may not be capable of creating left handed children, and the mutations that would create left handedness probably require several other changes to the world, plus the experiences I would have being left handed my entire life would shape me into a different person with different thoughts.

So the exact way the earth was assembled is a cause and effect relationship, without one you don't get the other. If you don't care what history the atoms of earth had before assembly then you can find infinite earths, but I argue that is not an exact copy.

Thank you for your thoughtful and respectful responses.