jmc

The maximum number of points at which a line can intersect 1 circle is 2 distinct points. Thus, for 3 circles, the maximum should be $3\times 2=6$ points at most. If you're going for speed, you should probably guess 6 points at this point with a reasonable degree of certainty. If you have time and want to be certain, you should only check for the existence of a line that intersects the three circles at $\boxed{6}$ distinct points, because it is impossible that a line could intersect the circles at more than 6 points. (There are, in fact, many lines that satisfy the conditions.)