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:
\[
S(n)=\min \left\{m \left| m \in \mathbb{N}^{*}, n \mid m!\right\} .
\]

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: There exist infinitely many composite numbers \( n \) such that \( S(n)=p \). Here, \( p \) is the largest prime factor of \( n \).
(1) Since $16=2^{4}$, we have $S(16)=6$.
From $2016=2^{5} \times 3^{2} \times 7$, we know $S(2016)=\max \left\{S\left(2^{5}\right), S\left(3^{2}\right), S(7)\right\}$.
And $S(7)=7, S\left(3^{2}\right)=6, S\left(2^{5}\right)=8$, so $S(2016)=8$.
(2) From $S(n)=7$, we know $n \mid 7$ !, thus $n \leq 7$ !.

Since $S(7!)=7$, the maximum value of the positive integer $n$ is $7!=5040$.
(3) For any odd prime $p$, taking $n=2 p$, we have $S(n)=p$.

Therefore, there exist infinitely many composite numbers $n$ such that $S(n)=p$, where $p$ is the largest prime of $n$.