jmc

We factor the quadratic, getting $(a-7)(a-3)\le 0$. The expression is equal to $0$ when $a=7\text{ or }3$. When $a\le 3$ or $a\ge 7$, the quadratic is positive, since the two factors have the same sign. When $3\le a\le 7$, the quadratic is non-positive. Therefore, $a=\boxed{7}$ is the greatest value of $a$ for which $a^2-10a+21\le 0$.