jmc

If $n\le 2007$, then $S(n)\le S(1999)=28$. If $n\le 28$, then $S(n)\le S(28)=10$. Therefore if $n$ satisfies the required condition it must also satisfy $$n\ge 2007-28-10=1969.$$ In addition, $n,S(n),\text{ and }S(S(n))$ all leave the same remainder when divided by 9. Because 2007 is a multiple of 9, it follows that $n,S(n),\text{ and }S(S(n))$ must all be multiples of 3. The required condition is satisfied by $\boxed{4}$ multiples of 3 between 1969 and 2007, namely 1977, 1980, 1983, and 2001.

Note: There appear to be many cases to check, that is, all the multiples of 3 between 1969 and 2007. However, for $1987\le n\le 1999$, we have $n+S(n)\ge 1990+19=2009$, so these numbers are eliminated. Thus we need only check 1971, 1974, 1977, 1980, 1983, 1986, 2001, and 2004.