jmc

The graph of $x^2+y^2=25$ is a circle centered at $(0,0)$ of radius $\sqrt{25}=5$. Beginning at $(-5,0)$ and working our way around the circle, we have the following 12 points on the circle:

$(-5,0)$, $(-4,3)$, $(-3,4)$, $(0,5)$, $(3,4)$, $(4,3)$, $(5,0)$, $(4,-3)$, $(3,-4)$, $(0,-5)$, $(-3,-4)$, $(-4,-3)$.

The greatest possible sum for any of these pairs is $3+4=\boxed{7}$.

(Of course, you could probably have guess-checked this answer relatively easily, but recognizing the equation as the graph of a circle is helpful in convincing yourself that there is no greater value of $x+y$ ... or, for example, if you wanted to find the greatest possible value of $x+y$, which is $5\sqrt{2}$).