The total area of all the gray rings is 2/3 the total area of the entire circle. Since the entire circle has a radius of 1, it's area is A = pi(1^2), or pi. The total gray area is therefore 2/3*pi, or (approximating pi as 3.14), 2.093.
The key is to compare rings, not circles. The outer gray ring is equal in area to the entire *circle that has a white ring. But half of that circle is white in area, and the rest is gray. So there is twice as much gray as white.
I tested this by creating a table of the area of each ring area (circle area minus area of the circle inside it. I then iterated this in Excel for 26 circles, alternating gray and white, and added the gray rings up, and added the white rings up. Of course, the series is actually infinite, but it was clear after these iterations that the ratio of total gray to total white approaches 2:1 as the number of circles increases. So an infinite set of circles would approach a ratio of 2:1. 2/3 of the total area is gray, and 1/3 white.
By the way, technically, a circle doesn't have an area. A circle is a line, all points equidistant from a center. The area is the area of the disc inside the circle. (http://en.wikipedia.org/wiki/Area_of_a_disc). It was a challenge keeping all my circles, discs, and rings straight. Or should I say round?
Calculus-Geometry, thanks for another great puzzle!