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Infinitely many.

Start with 1+665, 1+1+664, 1+1+1+663, ...

Then 1+2+663, 1+3+662, etc

and so on

Then you can start including negative numbers, and then fractions (rationals) and then irrationals. And, if you are up to it, you can even include complex numbers.

And that is only additions and subtractions.

So next you can look at multiplications and divisions and go through the same kind of routine.

And then you can try powers (exponents) and logarithms.

And then you can start combining these operations.

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Q: How many number combinations equal 666?
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