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.
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The number 1 is neither composite nor prime.
It is prime. By the way, any integer greater than 1 is either prime or composite. "Neither" is not an option in such a case.
Zero and one are neither composite nor prime.
Any number ending in a 5, other than the number 5 itself, is composite.
120 is a composite number. 120 is a composite number. A prime number has only 2 factors which are 1 and itself. Composite numbers are everything else except 1 and 0. 1 and 0 are neither prime, nor composite. Any even number (except 2) is composite.