Let $M\subset \Bbb{N}^*$ such that $|M|=2004.$
If no element of $M$ is equal to the sum of any two elements of $M,$
find the least value that the greatest element of $M$ can take.
1. Consider the set \( M = \{1, 3, 5, \ldots, 4007\} \). This set contains 2004 elements, and no element in this set is the sum of any two other elements in the set. This is because the sum of any two odd numbers is even, and all elements in the set are odd.

2. We need to show that the greatest element of \( M \) cannot be less than 4007. Assume, for contradiction, that the greatest element of \( M \) is less than 4007. Let \( M = \{a_1, a_2, \ldots, a_{2004}\} \) with \( a_1 < a_2 < \ldots < a_{2004} < 4007 \).

3. Consider the differences \( a_2 - a_1, a_3 - a_1, \ldots, a_{2004} - a_1 \). These differences must all be distinct and must be less than 4006 because \( a_{2004} < 4007 \).

4. Since \( a_k - a_1 \) must not equal any other element in \( M \), there must be 2003 distinct integers from 1 to 4006 that are not in \( M \). This is impossible because there are only 2003 elements in \( M \) and 4006 possible differences, which means there must be some overlap.

5. Therefore, the assumption that the greatest element of \( M \) is less than 4007 leads to a contradiction. Hence, the least value that the greatest element of \( M \) can take is 4007.

\(\blacksquare\)

The final answer is \( \boxed{ 4007 } \)