One common aspect of the various limit processes that have been described so far is that one has to take the limit of some quantity associated with each finite circle-it's not clear how to take the limit of the circles themselves if we insist on drawing our circles in the plane. But the plane plus a point at infinity is a sphere (think about the point at infinity at the north pole-hopefully I can get away with a bit of geography lingo here to simplify notation,) and this permits us make this problem finite, or compact. So for instance on the sphere the sequence of circles centered at the origin with radius increasing without bound converges to the north pole; a sequence running off to the right more quickly than its radius increases also converges to the north pole; a sequence centered at $(0,r)$ for radius $r$ converges to the prime meridian; and so on.
The answer still depends on a definition of "inside" and "outside" for circles on the sphere. Probably the most direct generalization of the same idea from the plane is that a point is inside a circle on the sphere iff it's at a lower "latitude," i.e. height, than every point of the circle. That makes the entire plane inside the limit of our first two example cases, but nothing inside the limit of our third, since the prime meridian includes the south pole. On the other hand, there's nothing outside the prime meridian either, since it includes the north pole! So this reformulation of the problem suggests that you need the ability to say "neither" in certain cases to answer completely.
By playing around with the third example it's easy to get sequences of circles which converge to any circle including the north pole. (The north pole has to be included since the radius went off to infinity.) So other than the latitude lines which include the south pole as well, these all have an "inside" but an empty "outside." This suggests that whenever the limit converges, the answer to your question is either "the inside is larger" or "neither is larger," and the former is in some sense more common. The fact that it's hard to come up with a natural sense in which the outside could remain larger, even though it's infinite at every time, is perhaps another reflection of the discontinuity of area from above that was mentioned in another answer.