jmc

The sum of the interior angles in an octagon will be $(8-2)\cdot 180=1080^{\circ }$. We know that the angles which are not right must have measure less than 180 if the polygon is to be convex. So, let $n$ equal the number of right angles in the octagon. The average measure of the remaining angles must be less than $180^{\circ }$, which is equivalent to: $$\frac{1080-90n}{8-n}<180$$ We can simplify this inequality: $$1080-90n<1440-180n$$ $$90n<360$$ $$n<4$$ So, the greatest possible number of right angles will be $\boxed{3}$.