Ogedei lives: How far can the Mongols get?

If Ogedei had not died in December 1241, how far could the Mongols get? The Holy Roman Empire? France? Italy?

There are numerous theories on that subject (and my list is far from being complete) so you can pick whichever you want:

1. Western - the Mongols got as far as they could because (a) further to the West they’d deal with the Western European who were superior (in all imaginable aspects) both to the Mongols and to their Easter European counterparts, (b) Western Europe had a lot of castles and (c) the Mongolian bows (and presumably the Mongols themselves) would fall apart as soon as the Danube is being crossed.

2. Russian - they would not go too far because they feared Russian rebellion at their rear.

3. All the way - they could go all the way to the Atlantic coast establishing an effective rule over Germany, France and Italy.

4. Great raid - this was just a Great looting expedition to fulfill Genghis’ order of going to the Last Western Sea. Within that theory the castles were not a big issue because they were not an obstacle to the great raid (however, as the later developments demonstrated, they could be a problem for the looting raids on a limited scale: the goodies were in the castles and you needed to take them to get a decent loot): during it the Mongols ignored all resisting places and kept advancing to preserve synchronization between their armies.

5. Not enough grass - it was computed by some specialist that Hungarian Plain presumably can’t support more than 30K horses (which were seemingly the only type of a livestock he heard about) and that as a result the Mongols could not use it as a base for the future operations. Obviously, there was no grass in the rest of Europe so the Mongolian expedition could not go further.

Keep in mind that a popular theory about requirement to drop everything and ride to Mongolia ASAP is a BS. Not only Batu did not get there at all but the main potential beneficiary, Quiuk, did not appear there for the next 2 or 3 years at which time the empire had been ruled by his mother as a regent.

Then, there was an argument that Quiuk and his supporters were not interested in expanding Batu’s domain, but at least some historians are objecting that this domain was officially extending only to the Kipchack steppes (and Russian lands) and that territories to the West would go to other Genghisides so perhaps Batu was not too interested as well. Not sure if territory that Nogay carved for himself in Bulgaria was vassal of Batu’s horde but Batu clearly had his hands (and pockets) full with what he already got and he was a cautious statesman, not a crazy conqueror for the conquest sake.

It is questionable if the Mongols could extend an effective lasting rule over the territories including Hungary, Poland and Germany because there was really no area nearby in which they could be present in the big numbers while maintaining a nomadic life style and still have enough troops to keep OTL lands under control: Within existing system Batu got only few thousands Mongols for his domain and there was ongoing conquest of China and Middle East. So in the best (for whom?) case scenario the Mongols would establish and maintain the same system of the vassal states as they did in Georgia, Russia and other places.

Interesting butterfly would be their invasion of Italy: they’d be killing opponents of Frederick II thus changing a balance in the Papacy/Imperial confrontation.
 
I think they would have gotten as far as Bulgaria the cost of the Adriatic parts of Italy and Germany
They did get to Bulgaria nd even established a Nogay horde there (for a while), their troops did reach Adriatic coast (which provided a formal excuse for wrapping up the 3xoedition) and they had been raiding parts of the HRE. The only thing missing is Italy.
 
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