I have approximated a lot. If it were product of reals, I believe my solution would have been fully justifiable. Let me know if my answer matches yours.

I will assume positive integers because or else we can maximize the product without bounds.

Lets say that there are y integers whose sum is 2008. Denote them by .

So

We want to maximize the product

Since by assumption the numbers are positive, we can apply AM-GM inequality.

Thus the maximum value of the product is and is purely determined by y.

So we want to maximize

A little bit of calculus establishes that f(y) approaches maxima at

Thus the discrete maximum is attained at (Actually I have done trial error to choose between 738 and 739)

In AM-GM inequality, equality holds when all the numbers are equal.

Thus

Thus I believe that is the largest product.