Casimir the Great (Kazimierz Wielki), King of Poland (1333-1370) had no sons. Well, no legal and aknowledged sons. But what if he had a legal heir to the throne?
In OTL Casimir was the last king of Piast dynasty. After his death Poland was ruled by his nephew Louis, king of Hungary, and later by Louis' daughter Hedwig (Jadwiga). She married great duke of Lithuania, Jogaila (Jagiełło) who converted to christianity and promised to baptism all Lithuania. It was the beginning of close relation between Poland and Lithuania which later transformed into union of those countries.
Polish noblemen decided to have Jogaila as their king because Poland and Lithuania had common enemy - the Teutonic Order. In 1410 Polish-Lithuanian forces crushed the Order's army at the battle of Tannenberg (Grunwald). Times of Jagiellonian dynasty are considered the Golden Age of Poland and Lithuania.
But what if Casimir the Great had a son? He was married 4 times (2 times as a bigamist!), had many lovers and nice pack of daughters, so he wasn'r sterile.
Let's assume he had a son with his second wife, Adelaide of Hess.
After his father's death 25-years old prince Władysław is crowned as new King of Poland. His land is in good condition: the chest is full, administration is well organized, Polish knights are good fighters, there's an academy in Cracow and Poland is widely respected in Europe. Problems? Teutonic Order on northern border, Silesia under Bohemian control and pagan Lithuania.
I think Władysław would try to baptism Lithuania (peacefully, of course) with one stroke gaining an ally against the Order and leaving Teutonic Knights without job. After all, they were supposed to fight pagans, right?
That is pretty much what happened in OTL. And then Polish king could try to get back Danzig (Gdańsk). But he wouldn't send his knights to fight. He would follow his father's example and send his lawyers to Pope's court.
Also, Piast dynasty was much more interested in Silesia and Pomerania, than Jagiellons, who rather looked east. So Władysław starts new campaign (not necessarily military one) to get those lands back to Poland. It was possible - Pomeranian princes were affraid of Brandenburg and Order, so they would be quite interested in becoming Polish vassals. Silesia would be a bigger problem, because Bohemian kings of Luxembourg dynasty considered it to be their land.
Shortly before his death Casimir was preparing war for Silesia. His son could continue his policy and about 1375 we have war between Poland and Bohemia. The Teutonic Order would probably try to interfere (on Bohemian side), but diplomatic action might be able to keep them out of war, and even if not, with Lithuanian support Poland could keep them at bay.
Poland wins the war (yes, I'm Polish). Silesia comes back to Poland: some of Silesians duchies are incorporated, some of them remain Polish vassals. Western Pomerania is happy to be Polish vassal as well, if only to have protection from Brandenburg (which also belonged to Luxembourg dynasty).
After defeating Bohemians, king Władysław the Brave turns against the Teutonic Order. Together with Lithuanians he defeats the Order in three battles and recaptures Danzig together with Eastern Pomerania. Unfortuately it costs him a part of Halicz Ruthenia - he must give it to Hungary in return for Hungarian help against Bohemia. But he is not to worried about: as I said before, Piast dynasty was not particularly interested in marching east.
King Władysław dies in 1420. He leaves to his successor much bigger and stronger Poland. He also didn't have to gain support of Polish nobles by giving them special privileges (in OTL Jagiełło assured in this way Polish throne to his son), so king's power is noticeably stronger. The Teutonic Orders still exists, but is too weak to cause serious problems. Christian Lithuania is only allied, not federated with Poland. Polish kingdom is now considered a real power.
Poland is more west-oriented and has stronger central power. Lithuania is now European first line of defence against the Tatars and later Turks (assuming Lithuania conquered Ruthenia). Poland of course supports its eastern neighbour, but it doesn't exhaust its resources.
There are no Hussite Wars, because it was Polish king who gave Hus guarantee of safety in Constanz in 1414. Polish knights, led by Zawisza the Black, protected Hus and didn't allow him to be arrested.
Hus starts a great movement of sanation of the Church. It takes a long time, but later, in XVIII century, he is canonized as a saint of Catholic Church.
Well, that's all for now. I run out of ideas. Well, not exactly...How about: Poles discover America?
