Genghis Khan lives to the age of 80 giving him 15 more years of Empire expansion as his generals no longer have to gallop back to Mongolia. How much more of the rest of the world does the Mongolian Empire include? Do the Mongolians ever learn how to conduct a naval invasion?
At best, they manage to conquer more of Hungary Austria. In reality, they probably would've just gotten bogged down fighting the Mamluks.
Okay, another interesting possible topic is, what would have happened if the Normans were defeated at Hasting.
England probably would've flipped back to the Danes before too long. That or Hardrada's descendents would take another stab at it and be successful. Like MonkeyHead said, they probably would've been inducted into the Scandinavian circle, and continue absorbing Norse culture instead of veering off and absorbing Norman French. I actually wrote a paper on the influence of Norse culture in England last year for a course (I'm a history major), and by the time the Normans came the English had already adopted a large number of Scandinavian practices and linguistic elements. It's not inconceivable to see them becoming part of a Scandinavian union, some successor to Cnut's "North Sea Empire" or whatever you want to call it. The ramifications for European politics would be huge though. First off, there would be no Hundred Years War and the English presence in France just would be less of a thing. Which would also probably mean an extremely reduced English presence in the Crusades, notably the 3rd, which was mostly an English endeavor. What would be interesting is the colonization of the Americas. I feel that Scandinavia would largely supplant England, and possibly even beat out the French earlier, the Scandinavians having a longer tradition of seafaring explorers. Add in the butterfly effect multiplying over the course of a millenium, and we'd have a very very different world today. That is assuming the Scandinavians won the inevitable infighting, which I think is the most likely outcome.