"They could have had the world so very easily."
Since there was insufficient pasture to be had in northern Italy or the lands beyond, The Mongols had to turn back, or their horses would perish. The invasion of Italy was not a rational option. Even after centuries of disafforestation and the conversion of arable to paddock in modern times, the Po Valley still only contains 4,247 square miles of pasture, barely sufficient to graze 38,925 steppe horses. Even if Batu and Sübodei had been willing to compromise their speed and mobility on such a narrow front by restricting the number of remounts to 5, the order of battle adopted for the Mongol campaigns in the Syrian desert, this would still only permit the ingress of 7,785 Mongol and Tatar cavalry. Thats less than 8 minghans or a single tüman, and Batu was equipped with only tactical artillery, no heavy siege train to facilitate the investment of major fortifications. The estimated population of the North Italian Plain was 3,218,000, of whom 603,000 lived in walled towns and cities, most likely defended by between 60,337 and 80,450 local troops, not counting mercenaries, of which there were plenty. Thats more than enough to cut off and obliterate the entire nomad salient, even assuming no reinforcements arrived in time from central Italy or France. And siege would take many months. The only way to occupy all of Europe is for mongols to adopt sedenmentary army which would take decades to change as the magyars and huns before them have done. While their disastrous campaign against Vietnam, Champa, Java, and Japan was all clear evidence for their limit of power.
"Rome was the most powerful. We will have to see if the US manages to be number one for the next centuries as was Rome in its day leading to a Pax Romana which lasted more than 8 centuries."
reality check. Rome was never number one.
"Up until about the 1900's I would have said Britain, since it at one time controlled just under 1/4 of the earth's surface and just over 1/4 of the population."
Yet despite its territorial height, it was already losing its edge, for its navy was challenged, while its army never had the continental edge as the continental powerhouses exercised. Just been picky, the population of the world in 1900 is roughly 1.62 billion, Britain's subject number some 372 million souls, less than a forth of the world's population.