What about an accidental World War? Its not unlike what Koprulu Mustafa Pasha proposed, but in the Middle Ages. Picture a TL in which the Mongols are even more successful, invading India, central Europe and managing to disimbark into Japan, into the same exact timeframe (let's say, during Ogedei's later years, before the Hordes are partitioned), well beyond their OTL conquests. The Pope, in a proto-Crusade of Nicopolis, calls a Crusade against the Mongols in Europe, and most of the European feudal polities (barring Spain, perhaps) - France, the HRE, some Italian duchies, Hungary, Poland and perhaps Byzantium, if they go through the Balkans, or the Russian principalities rising in rebellion - join the effort, in the exact same time the Japanese and Chinese from southern Han are facing their armies, and the northwestern Indian states are trying to oppose them (perhaps the war effort could join the predecessor state of Delhi, Gujarat, Sindh and even Bengal), and the Mamluks in the near East.
This is somewhat like the patterns of invasion that happened IOTL, but here I'm thinking about a brief and specific timeframe (some 3 or 5 years) in which these wars happen simultaneously instead of along various decades of invasions - the Mongols would be coordinating war theathers from Europe to the Near East and the Indian Subcontinent, going to the Far East, and, considering their quasi-ASB record of successes, one can imagine that they would win and counterattack in some of these scenarios, expanding even further - how about Mongol raids into France or a Mongol vassal in Japan or India? The very fact that we can conceive this scenario is a testament to how truly global the Mongol Empire was, and, from their perspective, any of these unrelated and distant theaters of war could be regarded as a "war against the whole world", and would exclude only Africa (if we ignore the Mamluks of Egypt) and the Americas.