PoD-for-the-day (#9 - March 20th, 2007): Valdemar II wins the 1227 Battle of Bornhoved, defending his epithet "the Victorious" and subduing his rebellious German vassals with the help from his ally, Emperor Friedrich II. The Danish hegemony in southern Baltic is preserved, and Valdemar II doesn't quit his conquering career, instead campaigning in the Baltic. When the Livonian Brothers of the Sword are crippled by Lithuanian pagans in the Battle of Schaulen, they ask their sometimes-enemy, sometimes-ally Valdemar for help, which he provides in exchange for their feudal allegience. The Lithuanians are repulsed and Valdemar spends much of the rest of his life (he died in 1239 ITTL) campaigning in the east. Campaigns against Novgorod aren't very succesful, but Mindaugas and his Lithuanians are decimated, allowing the Dano-Livonians to methodically slaughter entire tribes over the next few years (ironically, the hardline measures taken by Mindaugas to pull the tribes together backfired, as after his death there were few other leaders left in good positions, and those were unable to coordinate their efforts to any effective extent).
The later Dane-kings were unable to conquer much more, although some gains were still made; they also lost a large degree of their power to the Danehof (a parliament; merchants are more important than in the OTL Danehof). By the 14th century Denmark was a somewhat stagnant empire, although its control over strategic trade routes made it one of the richest realms of Europe (however, the Baltic trade's development has been ******** as compared to OTL, as no Hanseatic League was allowed to rise; consequently, western and eastern Europe - more precisely, England, Flandres and Novgorod - are more important and prosperous than in OTL).
Sweden hadn't done much over the 13th century, apart from consolidating control over Finland and launching some bungled attacks against Novgorod; of somewhat more importance were the machinations that allowed the early 14th century personal union with Norway, which had split away from Denmark in mid-13th century. Predominant in the northern Baltic and partaking in North Sea trade, Sweden is doing quite well.
In Central Europe, things had gone quite differently, with the Holy Roman Empire and the House of Hohenstaufen faring very well indeed, thanks to its Danish and Teutonic allies who had provided it with invaluable military assistance in time of need. Conradin had succesfully defended his crown and his Empire, even campaigning in the Middle East, without serious losses or gains for his "Kingdom of Jerusalem". The Empire had begun eroding under his successors and the Kingdom of Jerusalem was lost to the Muslims, buton the other hand the allegience of the Teutonic Order, how ever nominal, was acquired and Poland - devastated by the Mongols and kicked while its down by the more southwards-oriented Teutonic Order - under partial colonisation and integration, much like Bohemia. The Holy Roman Empire in the early 14th century was a colossus with clay feet, but a hegemonic realm nonetheless. As for the Papcy, it was in crisis as a German puppet Pope sat in Rome and a French puppet Pope sat in Avignon. A "cold war" of sorts was ongoing, and yet another factor in favour of the Holy Roman Empire was the flight of the Templars from France to Swabia, where they found refuge and support, which they were able to reward heftily.
France was weaker; though the English were limited to Guyenne and Normandy, and the Imperials failed to make any lasting gains during the on-again off-again wars along the border, the Flemish rebels in the late 13th/early 14th century had humiliated the French knights repeatedly and defended their independence, quickly becoming a small, yet prosperous merchant republic. On the other hand, due to different dynastic developments Provence was united with France earlier.
England may not be much stronger than in OTL at the first glance, although it does have control over Normandy and Scotland; however, it has great economic strenght and potential, the trade is booming and Templar banking is in development. A war with France is ongoing, however, and it tends to consume some resources.
Aragon may have been thwarted in its Mediterranean ambitions by the Hohenstaufens and the failure to organise a proper coalition, but it remained a great economic power, and a significant player in the Imperial anti-French coalition. It had managed to form a personal union with Castile, but not with Leon (which succesfully rebelled); it conquered Navarre, though, and more recently Granada and a few North African cities and castles.
Hungary lingered on under Late Arpads, losing much power and influence, as well as periphereal territories, in the aftermath of the Mongol invasion. However, something of a revival occured in the late 13th/early 14th century, as Andreas III of Hungary inherited the throne of the increasingly-Catholicised Russian principality of Galicia-Volhynia. The Arpad dynastic empire is very unstable, though, both due to legal and religious differences and because the dynasty is on the brink of extinction.
The Paleologid Byzantine Empire and its Genoese allies had managed to destroy most Latin states; a policy of cooperation with the Holy Roman Empire even allowed for a brief renaissance and reconquest of some of the Anatolian territories, but as of the 14th century the Second Rome was once more filled with strife and intrigue, slipping back into decay. As for the Balkans, they were mostly dominated by an ascendant Serbian kingdom, though Bulgaria, a grudging Byzantine ally, managed to block the Serbian ascendancy somewhat.
The Russian princes, still under the Mongol knout, retained a great degree of autonomy. The trading republic of Novgorod gained great economic influence, though it had failed to expand over the 13th century, apart from the occasional conquests of Pskov. The rest of Russia was dominated by the principalities of Moscow (in the northeast), Polotsk (in the west), Tver (in the southeast/centre) and Galicia-Volhynia (in the south); the latter is now in union with Hungary, however. Also there are many smaller principalities, such as the once-great principality of Kiev. Moscow may be the strongest militarily, but Tver and Polotsk are in a close alliance and are also more developed, leaving the good relations with the Mongols Moscow's only true advantage.