This seems easy - since the limit of the length is infinite, you could only get a fixed point as you go from level n to n+1 if there's some very nice (rational) relationship between the size of the ball and the length of the level n snowflake.

Here's another way to state Josh's point: "How can you roll a ball along a Koch curve when you can't even roll a *hoop* along a Koch curve?" (If we mark a point on the hoop, and roll it along a polygonal approximation to the curve starting with the marked point touching the plane, then when the hoop has finished its journey, the angle between the marked point, the center of the hoop, and the new point of contact between the hoop and the plane will be L/R mod 2 pi, where L is the length of the polygonal approximation and R is the radius of the hoop. But there's no reason to expect L/R modulo 2 pi to have any kind of good limiting behavior as we consider a sequence of polygonal approximations with L going to infinity. Indeed, by continuity, we can construct polygonal approximations that will make L/R mod 2 pi equal to any angle you like.) Jim Propp