The fastest way to get max population is to get rid of monogamous cultures (e.g. European) and replace them with polygamous cultures (e.g. Middle Eastern/Indian/Chinese). Other things that could help: more rulers/kings (so more harems) and conservative societies (so massive focus on family/procreation).
Wars/totalitarian regimes/medicine are probably not as important as one might like to think. China lost millions of people in the mid-17th century, but managed to recover in 150 years. Slavery cost 20 million over the course of three centuries; this deficit can be filled in within a couple generations if we have "max population" cultures settle in Africa/America.
My take:
1. 1400. Timur turns north (instead of towards the Ottomans OTL) and unites the Golden Horde under his banner. Seeking to best Genghis Khan, he careens his horde+cannons into Central Europe through Russia, exterminating/pastoralizing Poland, Germany, France and Hungary. Central Europe turns into a wasteland. (0.35bn OTL/0.3bn here)
2. 1402. Without Timur to stop them (as they were in OTL after the Battle of Ankara), Ottomans capture Constantinople and, claiming to be heirs of Rome, capture the city. This inaugurates a new jihadist spirit throughout the Islamic world, and Islam once again begins its terrifying expansion - Sub-Saharan Africa (Morocco/Mamluks), India (Delhi/Gujarat), Europe (Ottomans/Morocco/Granada). The last bastions of Christendom crumble before this renewed onslaught. (0.35bn OTL/0.3bn here)
4. 1405. With no Timurid horde to escape from, the Oirats and other Mongols are not pressured to invade Ming China. The Emperor Yongle can thus focus on his most pressing task - finding his nephew the Jianwen Emperor, whose throne he usurped in 1402. Tens of fleets are sent to scour the world's seas. Through them, China learns of the existence of the Philippines, Indonesia, Australia and Siberia. (0.35bn OTL/0.3bn here)
5. 1410. Yongle, still having not found Jianwen, continues rounding up and executing Jianwen-sympathizers. The numbers are in the hundreds of thousands and contain many of China's intellectual elite (same as OTL). Xie Jin daringly intercedes on behalf of the prisoners and argues for transportation to the newly-discovered lands is an alternative. Yongle agrees and China begins dumping prisoners in Australia and Siberia. His successors continue Yongle's maritime and settlement policy. (0.35bn OTL/0.275bn here)
6. 1500. Numerous jihadist states being set up along trade routes in Mali, Nigeria and Tanzania. Without Portugal to interrupt Arab shipping, Arab traders eventually round the Cape and colonize it. Meanwhile, Muslim Europe fractures and collapses after a series of internecine wars: city-states now abound in the region (just like in Moorish Spain OTL). Rulers soon figure out that more people = better odds versus their neighbors. Population in Muslim Europe is nearing pre-Timur levels and is continuing to increase, Muslim leaders drawing on Greek and Roman texts to build canals, aqueducts, etc. (0.425bn OTL, 0.4bn here)
7. 1600-1700. Ming penal colonies become fully fledged states in their own right, the sheer abundance of land in these virgin lands acting as a major pressure valve for stability as dispossessed farmers simply move outwards in search of new land. Similarly, excess population in Islamic Europe is 'drained away' by continued colonization at the frontiers: Scandinavia, Scotland, Russia, even in Brazil and Southern Africa. Despite the constant wars, this period is relatively stable and will be known for its cultural and scientific achievements (probably equivalent to the Renaissance). (0.545bn OTL, 0.625bn here)
8. 1700. Ming Chinese colonize California and begin mining its gold and silver. This proves to be disaster for the world as the influx of precious metal destabilizes the world, disrupting the Ming and Ottoman Empires, who both totter and collapse spectacularly. Known as a "Global Thirty Years' War", Old World nations fall on each other and create a bloodbath not seen since the days of Timur. Desperate peoples fleeing war expand Muslim/Chinese enclaves in America, its native inhabitants dying from smallpox. In addition, the bleakness of this period generates millennarian movements throughout Islam once again, and gazis conquer numerous tribes/nations in the Dar al-Harb (House of War): Kongo in Africa, the Aztecs/Inca in America, perhaps Burma/Tibet in Asia. On the other side of the world, Chinese settlement crosses the Rockies and pours into the Great Plains. (0.6bn OTL, 1.0bn here)
9. 1800s. Utterly exhausted with war, Islamic and Chinese societies move to more 'regulated' forms of combat (like how European wars in the 1700s became much less bloody than in the 1600s). This an age of first recovery and then prosperity - accelerating further as increasing trade links result in greater production, which eventually sparks off an Industrial Revolution in the 1850s, Western Europe and central China (where the major Song steelworks were) being the first places to boom. (0.9bn OTL, 1.8bn here)
10. 1900s. Massive progress in science, medicine and agriculture. These are not followed by corresponding advances in belief, however: the Islamic world takes these developments as sign of God's power (which also offers people relief from 'technological shock') and China retains its age-old belief in familial piety, which technology does nothing to erode. In both cultures, the constant presence of harems continues to devalue women and while a few strong-willed ones do break from the mould, the majority remain comfortable being family matriarchs. Nationalism never occurs; and so wars remain frequent and small-scale as the political world remains deeply fractured. Any remaining areas uncolonized (Canada, Amazonia, Congo and Siberia) are taken over. (1.6bn OTL, 3.0bn here)
11. 2000: (6.0bn OTL, 11.0bn here)
12. 2013: (7.0bn OTL, 12.0bn here) - I think a population in the low-teens is probably reasonable as a max for human population on Earth (esp. an industrialized population, though in this world I doubt consumerism would exist) - any more than that and Mother Nature will quickly slap you back down. Life in this world will be pretty bad, I suspect, especially if you are a woman.
Hopefully I didn't alter human psychology too much (war still happens,for example).