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For me, it's the Liouville numbers. They are a special type of transcendental number which can be more efficiently approximated by rational numbers than any other irrational number, including algebraic irrationals. This is counterintuitive because we see rational and algebraic irrational numbers as being closer to each other (due to both being algebraic) than transcendental numbers.
It's like meeting your distant third cousin, and finding out they resemble you more than your own sibling.
(Flairing as "number theory" because I had to make a choice, but the question applies to all fields of math.)
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