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There are infinitely many such numbers.

For example, any irrational number between 0 and 1. The principal square root of any number between 0 and 1 which is not a ratio of perfect squares.

Such a number will have infinitely many digits, and lie between 0 and 1.

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Q: What is a number with a lot of digits that is smaller than 1 but larger than 0?
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