14. In 1993, American mathematician F. Smarandache proposed many number theory problems, attracting the attention of scholars both at home and abroad. One of these is the famous Smarandache function. The Smarandache function of a positive integer \( n \) is defined as
\[
S(n)=\min \left\{m \left| m \in \mathbf{Z}_{+}, n \right| m!\right\},
\]
For example, \( S(2)=2, S(3)=3, S(6)=3 \).
(1) Find the values of \( S(16) \) and \( S(2016) \);
(2) If \( S(n)=7 \), find the maximum value of the positive integer \( n \);
(3) Prove that there are infinitely many composite numbers \( n \) such that \( S(n)=p \), where \( p \) is the largest prime factor of \( n \).
14. (1) It is easy to know that $16=2^{4}$. Then $S(16)=6$.

From $2016=2^{5} \times 3^{2} \times 7$, we know
$S(2016)=\max \left\{S\left(2^{5}\right), S\left(3^{2}\right), S(7)\right\}$.
Also, $S(7)=7, S\left(3^{2}\right)=6, S\left(2^{5}\right)=8$, so $S(2016)=8$.
(2) From $S(n)=7$, we know $n \mid 7!$.

Thus, $n \leqslant 7$.
Notice that, $S(7!)=7$.
Therefore, the maximum value of the positive integer $n$ is $7!=5040$.
(3) It is easy to know that, for any odd prime $p$, taking $n=2 p$, we always have $S(n)=p$.
Since there are infinitely many odd primes, there exist infinitely many composite numbers $n$ such that $S(n)=p$, where $p$ is the largest prime factor of $n$.