$ f(g(x)) $ must have a degree of 8, because it will produce the term with the greatest exponent of the polynomial. Because $ f(x) $ is a degree 4 polynomial, we can write that $ f(x)=bx^4+cx^3+dx^2+ex+f $. The term with the greatest exponent in $ f(g(x)) $ comes from taking $ bx^4 $ or $ b(g(x))^4 $. Let $ g(x)=a_{n}x^{n}+a_{n-1}x^{n-1}+...+a_{2}x^{2}+a_{1}x^{1}+a_0 $. Then, the highest degree term of $ f(g(x)) $ is $ b(a_nx^n)^4 $, which equals $ ba_{n}^4x^{4n} $. Since the degree of $ h $ is 8, we have $ 4n=8 $, so $ n=2 $. Therefore, the degree of $ g $ is $ \boxed{2} $.