The number 1.
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Every positive integer greater than 1 is either prime or composite.
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.
A prime number has exactly two factors, 1 and the number itself. Any number having more factors than a prime is a composite number.
All positive whole numbers over 1 can be classified either as prime or composite numbers. Those which are not prime are composite numbers, that is, numbers which are composed of several different factors.