jmc

The minimum possible degree is $0$, since we can find polynomials such that $f(x)=-g(x)+c,$ where $c$ is some nonzero constant. This gives us $f(x)+g(x)=c,$ which has degree $0$. Since we are looking for the product of the minimum and the maximum possible degrees, we can easily see that our answer is $\boxed{0}.$

Note: The maximum possible degree of $f(x)+g(x)$ is $7,$ since it is impossible for a sum of two polynomials of degree $7$ to include any terms of degree higher than $7.$