jmc

We see that $-4$ is not in the range of $f(x)=x^2+bx+12$ if and only if the equation $x^2+bx+12=-4$ has no real roots. We can re-write this equation as $x^2+bx+16=0$. The discriminant of this quadratic is $b^2-4\cdot 16=b^2-64$. The quadratic has no real roots if and only if the discriminant is negative, so $b^2-64<0$, or $b^2<64$. The greatest integer $b$ that satisfies this inequality is $b=\boxed{7}$.