I considered deleting the post, but this seems more cowardly than just admitting I was wrong. But TIL something!
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You're right that they're the same size but you're mistaken when you try to assign a total value to the stack. Consider breaking each $100 bill into 100 $1 bills. The value is the same, clearly. So for each pair, you have a $1 bill and a small stack of 100 $1 bills. Now combine all singles back together in an infinite stack. Then combine all stacks of 100 into an infinite stack.
And you know what? Both infinite stacks are identical. They have the same value.