jmc

$f(g(x))$ must have a degree of 6, because it will produce the term with the greatest exponent of the polynomial. Because $f(x)$ is a degree 2 polynomial, we can write that $f(x)=b{x}^2+cx+d$. The term with the greatest exponent in $f(g(x))$ comes from taking $b{x}^2$ or $b(g(x){)}^2$. 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 $b(a_n{x}^n{)}^2$, which equals $b{a}_n^2{x}^{2n}$. Since the degree of $h$ is 6, we have $2n=6$, so $n=3$. Therefore, the degree of $g$ is $\boxed{3}$.