dt = 0.693*sqrt[Tl*Tf*Wf /(1?Fi)]
where Tl is the time to build one labor droid in seconds, Tf is the time to build one factory in days, Wf is the number of labor droids required to build one factory, Fi is the percentage of the factories output that is infantry droids, and dt is the doubling time in seconds. The result is very low; if it takes 10,000 labor droids 30 days to build a factory, and the factory line produces droids at a rate of 1 per second, then it will take 12 days for the Confederate Droid Army to double in size. Using the doubling time, one must now simply plug that value into the exponential growth equation to derive the might of the Confederate Droid Army.
It is this strength more then any other that defines the scope of the Clone Wars. However, it also begs the question as to why they went on as long as they did – an independent commander could increase his labor droid production to shorten the doubling time and thus provide himself with the mean to win single handedly. He can do this free of the influence of Lord Sidious, for while the Dark Lord is powerful he does not control everything, and his failures have had dire consequences (The Phantom Menace, Labyrinth of Evil). Exponential growth is not simply limited to ground forces, it also follows to materials, ships, and supplies (Destiny’s Way). Materials to construct the droids do not need to be shipped across the galaxy, as droids are made of common materials in planetary systems (Star Wars: Complete Locations) thus explaining why the Republic observed so little material trafficking (Odds). Factories can be hidden anywhere; being operated by droids they are not confined to habitable worlds or even systems – an asteroid belt around a brown dwarf would do. Single factories are self-replicating (Revenge of the Sith Incredible Cross Sections), and can swiftly cover a world, turning it into one the mentioned “Factory planets” (Star Wars Databank – Techno Union). Such factories were already spread across the galaxy (Star Wars Databank - Geonosis Droid Factory; Star Wars Databank – Geonosian; Star Wars Tales 6). The term “factory planet” is literal – the surface of a planet is covered in factories, mines, and starports to the exclusion or everything else (The Essential Guide to Planets and Moons). Consider the production of a single one of these planets – Hypori is notable for its revelation. An earth sized planet, even if similarly covered by water, with a mere 10% ofits land used for factories one square kilometer in size with a production capacity equal to that seen in Poggle the Lesser’s hive will produce four trillion (4*10^12) droids each day. It is notable that the production lines are outdated (Star Wars: Complete Locations), and that other dedicated factories would be both bigger and faster. It is thus in the best interests of the Confederacy to co-opt any factories it can find (General Grievous), and the Republic must constantly be monitoring all Confederate activity to find and eliminate any factories they construct or conquer (Republic Commando: Triple Zero; Clone Wars: The Best Blades).
Why then, did it take so long for the Confederacy to out produce the Republic? The growth time for clones means that the Confederate Droid Army will have a much shorter doubling time then their Republic counterpart. The answer is found in their equipment and their tactics. The factories of the Confederacy would need to produce more then just infantry droids; the vehicles and artillery used by the Confederacy would surely require more time. However, the time to expand and retool to produce another type of droid is negligible on a galactic scale. The Republic was attacking and sabotaging factories (Republic Commando: Triple Zero; Odds; Clone Wars Chapter 20; Clone Wars: On the Fields of Battle) as fast as they could find them, but again that would be negligible. The biggest clue is the mining of hyperspace lanes (CIS Shadowfeed 14:7:01, article "Kuat-Neimoidian Arms Race Escalates" SW Insider 68, page 69). Space is so vast that any real attempt to mine it is futile, but Republic space superiority demanded the Confederacy find a way to counter it or risk destruction from orbit. Only exponential growth and a significant amount of time allows for mining of such a vast area, thus this would use the bulk of the Confederate manufacturing base for almost the entire war. This focus would explain the low numbers of droids observed being produced in some of their most extensive factories (Odds) as those factories would presumably only be producing a small amount of labor droids to increase production in favor of mines. It would also explain why it took the Confederate Droid Army so long to overtake the Grand Army of the Republic in size. (Dark Rendezvous) Once in full swing though, even directing an inconceivably small fraction of this required output would drown out the civilian population of the galaxy in short order.