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Yes. All prime numbers and composite numbers are positive integers, or whole counting numbers. That leaves infinitely many numbers that are neither prime nor composite. If you intended to narrow the scope of your question to the whole counting numbers or to the positive integers, then there are NO such numbers that are neither. A counting number, however large, will be either prime or composite.
It is 1 that is neither a prime or a composite number The number 1 is neither prime nor composite.
There is no special name for it. It is simply neither prime nor composite.
1 is neither prime nor composite.
It is not a prime nor a coposite. It is a composite number, though.