Doubling time is important if movement are to seep into populations. But movements don't "cleanly double." Movement multiplication is messy: some people make disciples and many people don't.
Pareto's Law, which shows up in most human endeavors, states that "for many events, roughly 80% of the effects come from 20% of the causes." We see this in everything from economics to business to giving in church to the number of people who make evangelists. We will see it in a movement, as well: most of the multiplication will come from a minority of the groups.
Below is a little theoretical spreadsheet. Starting with just 10 believers, we apply the 80/20 rule. Column 1 is the number of "iterations" (could be months or years). Column 2 is the number of believers. Column 3: 80% of those believers do nothing - they make no converts or disciples. Column 4: 20% make just 1 disciple. Column 5: 20% (of column 4) disciple a family (maybe their own; we average this to 3 believers). Column 6: 20% (of column 5) disciple multiple families (in this case, we estimate 4 (or 12 believers). Column 7 is the total number of new believers added by these 20% efforts (with the T columns showing the total believers from each of the 20% efforts).
It is amazing what just 20% of a group can do! In 35 iterations, this movement would completely saturate a 100,000 population province. (By iteration 27, they would certainly be noticed.) In 43 iterations, they could significantly impact a megacity (perhaps even saturate it). Although it would be ideal to have 100% cooperation, you do not have to have it. Normal human activity levels (80/20) would be sufficient.
Again, how long each iteration takes is important. If it takes a year before someone can make a disciple, this chart would extend out in time very far indeed.
A simple fluctuation in the % level can make a huge difference. What if a given church only has 10% making disciples, rather than 20%? Then we'd have a chart like the one below, representative (unfortunately) of a lot more churches.
On the other hand, even a small fluctuation up can have dramatic results. If 25% are active, you would saturate a 100 million population in 43 iterations.
Neither of these charts take into account births or deaths (which generally only sustain a Christian community) nor immigration or emigration (which can dramatically impact small communities under oppression). But they both show that shepherding a small community through the critical early 19 iterations is vital.