Some time ago I proved that in the long run, after playing for a time t, cookie production is proportional to the square root of t and total cookies is proportional to t^(3/2). This results from the fact that CpS is proportional to prestige, which itself increases as the cube root of the total number of cookies.
I did this before the introduction of the Birthday Cookie. BC adds an additional 1% to the CpS for every year elapsed. This means that in the VERY long term (we are talking about decades and centuries here), production eventually becomes proportional to t^2, and total cookies becomes proportional to t^3.
All this of course assumes that people will still be playing Cookie Clicker in 100 years' time. If they are, then I imagine Orteil's grandchildren will have implemented many fixes and improvements, one of which will probably be to limit the maximum benefit from Birthday Cookie. Or maybe they won't. When your number of cookies is in the quinquidecillions, a factor of 10 here or there may not be considered worth bothering about.
You can see the effect of Birthday Cookie in this graph. Both axes are on a logarithmic scale. In the early years, the slope is close to 1/2, reflecting the fact that production varies as the square root of time. At about 30 years the effect of BC becomes noticeable, and after several hundred years the slope approaches 2, indicating that CpS is proprtional to t^2.