Special number/Exercise

A natural number is called special if it fulfills a specific explicit property. $0$ is special, as the neutral element of the addition, and $1$ is special, as the neutral element of the multiplication. $2$ is the first prime number, $3$ is the smallest odd prime number, $4$ is the first true square number, $5$ is the number of fingers of one hand, $6$ is the smallest number composite of different prime numbers, $7$ is the number of dwarfs in the fairy tale, etc., so these numbers are all special. Does there exist a number which is not special? Does there exist a smallest number which is not special?

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