Beginning as OTL
Being too far away from OTL's Mongol conquests, French history took a while to divert significantly. From 1209-29, France fought and eradicated the sect of the Albigensians, acquiring Toulouse for the king.
At the same time around they had to fight the Angevine Empire; although France was initially weaker, the former ended with the Battle of Bouvines in 1214. This victory helped Philippe Augustus to make France a hereditary monarchy in 1226. Later in 1259 England lost all possessions in France but Guyenne.
Louis IX was a very pious ruler; in 1263, he planned a crusade as retail for the lost cities in Palestine, but his brother Charles of Anjou was busy in Italy, so it was postponed. In 1270 finally, the Seventh Crusade happened. France attacked Tunis (which was wanted by Charles too), without success.
In 1276, Philippe III of France fought Castille for reasons of succession, without success. In 1296, the Auld Alliance between France and Scotland was made.
In the north, France had tried to get Flanders under its control. But in 1302, Flemish citizens massacred French occupation troops, and defeated them in the battle of Kortrijk. The fights went on, until Flanders and France made peace in 1312; Flanders ceded its French-speaking parts.
In 1303, Philippe the Fair was excommunicated by the pope. In the same year however, Pope Boniface VIII was seized by the French. In the following decades, the (mostly French) popes would reign from Avignon, which was chosen as the new residence in 1309.
1306, the Jews were kicked out of France - the state was almost broke, and the king wanted to take their money. For the same reason, the Knights Templar were accused for heresy in 1307 and again in 1311/12, which ended with their destruction.
Divergence and First Aquitainian War
After the deaths of Philippe IV the Fair 1314 and his son Louis X (who may or may not be identical to his OTL counterpart) in 1317, the latter's baby son Jean I became new king. At the beginning, his uncles Charles (-1323) and Philippe (-1327) reigned in his name. In 1336, the young king started a short war with Genoa.
Although the question of succession didn't become as urgent as OTL, where it lead to the Hundred-Year War, England and France still managed to find an opportunity to fight. In 1341, the Breton War of Succession began. France supported Charles of Blois, England John of Montfort - with the result that both were drawn into the war, which now became the First Aquitainian War. It was joined by Flanders on England's side and Scotland on France's side. Thanks to their longbows, the English defeated the French several times; and although the Black Death wouldn't happen until some decades later, France was hit very hard by the famines and the Peasants' revolt of 1351. At least, in 1349 France acquired the Dauphine.
In 1353, France has to make peace with England after the defeat in the battle of Caen. John of Montfort became duke of Brittany, England got all of Gascony (similar as in the peace of OTL 1360), Flanders acquired the Artois.
Now the French king thought about his family and planned to give several duchies from the crown's domain to his sons, as appanage. If that plan had been realized, the French kings would've lost half of their power, since he had no less than four surviving sons. However, in 1355 the French queen Isabelle deposed her incompetent husband and made her son Jean II king. About the other sons, the queen promised them lands that had yet to be conquered, in the English territory.
In 1366, Burgundy went to France. In the same year, the French started the war against England again. Since they had learned how devastating the English / Welsh longbows were, they avoided open fight - with success. In 1385, England and France make peace again; the English lost what they conquered in the last war, going back to the borders of 1340.
1396, the Black Death arrived in France.
Second and Third Aquitainian War
1414, England under new king Richard III made alliances with Castille and the Netherlands, attacked France again. The Second Aquitainian War had begun. While France suffered some defeats in the beginning, 1421, they made a separate peace with the Netherlands, giving emperor Gerhard II Flanders, which was incorporated into the Netherlands. Now, the tides turned again: Storming the conquered cities with cannons and relying on their greater manpower, the French slowly drove the allied English and Castillians back. 1424 the Cortes, the Castillian estates, protested the high costs and low results of the war. King Pedro III was sorry about it, but had to leave the war too. Things were complicated again however when the pope and the Anjou king of Naples asked the French king in 1426 for a crusade against the Rum-Seljuks. King Charles IV agreed, made peace with the English, leaving Guyenne in their hands; king Richard III had to accept it as a fief, though.
1439, Brittany got a pro-French ruler, eventually returning to the fold.
But in 1455, France was shook up again when king Louis XII had his completely mad younger brother Philippe (also duke of Bourbon) killed. The nobles (including the royal sidelines, and old king Richard III of England, who was talked by his advisors into it) who were already concerned with the growing power of the king, used this accident to rebel against him. 1460, the first French Civil War ended, Louis' other brother Charles V became king. He had to grant the other dukes a lot of power, however, which made them practically independent.
Now however he was free to make the Third Aquitainian War 1477-83, which ended with Aquitaine finally becoming French. Scotland-Norway also had entered the war on France's side, but this ended with Scotland falling under English rule.
1486, France sent the first ships to Atlantis, under a captain Coulon (of the infamous pirate family), making claim in the area of OTL Carolina, where they founded Charlesbourg at the site of OTL Charleston.
In 1490, Charles VI made the Republic of Venice appoint his cousin Jean "protector of the republic", essentially giving him lots of real power, promising them an "everlasting alliance" with France against the Muslims. This alliance would be expanded: In 1511, François, the Regent for his nephew, French king Charles VIII, was elected Roman king (as Franz I). He had to find out soon that the title was far from meaning much power. To counter this, he founded the "Alliance des Alpes / Alpenbund" with Venice, Bavaria and Württemberg-Carinthia, which is mainly directed against the Seljuks.
1522, the king was warned by Prince Alasdair that the Quadruple Monarchy was about to conquer the rich lands of the Tenochca. Alasdair pointed out: "If the king owns their gold, he can afford to hire all the mercenaries in the world! You have to stop him now!" King Charles agreed with him, started the Great Occidental War. French troops fought in Iberia, North Africa and Atlantis, and in the peace of Barcelona, France acquired West Florida, Panama and the Mississippi valley and, maybe more important, became the protector of the native states of Mexico and Tawantinsuyu (the Inca empire).
Now that the Occidental Schism was ended and the main opponent weakened, France turned its look eastward. 1567-74, they fought the first French-Seljuk War against the weakened Seljuks. In the peace of Krems (Austria) 1574, the Seljuks had to cede western Hungary to France. François of the royal sideline of Bourbon was elected new king Ferenc I of Hungary.
The 1570s became "The golden years": All the great Western European states - the Triple Monarchy of England-Castille-Portugal, France, and Alasdair's empire had competent, popular rulers who lived in peace with each other. The Style of Occidentalism(o) was at its height.
After the sons of Charles VIII had died without heirs, their second-grade cousin François of the Berry sideline became new king François II of France in 1586. He was anything but popular, but fortunately, he'd die one year later already.
1599, the kings Henry V of the Triple Monarchy of England-Castille-Portugal and François III agreed to solve their conflict in Aragon in a way that satisfied them both: Aragon was divided - Murcia and the lands south of the Ebro went to Castille, the lands north of it, plus the Baleares, Navarre and Sardinia (which was sold to Florence, though) to France. They also fought the Great Seljuk War together, further strengthening their empires.
The new king had to cut down the power of the nobles at the court (1619-25), finally succeeding and making France an absolutist state. About this time, the formerly good relations between the Triple Monarchy and France cooled down dangerously, which was also the fault of the other new monarch, Henry VI. It was a triumph for François IV when his opponent first lost the Iberian peninsula, and later even the crown of England. So it was no wonder that France supported uprisings in Castille 1628 - and when the Castillians had been successful, France also supported uprisings in Aragon and Portugal, to weaken its neighbor further. In 1629, he used the opportunity to grab Algeria, where the Castillian rule wasn't firm enough yet. In Atlantis, the French-Spanish border between Caroline and Florida was moved south, giving Virginia (OTL Georgia - named after the Virgin Mary) to France.
He lead France into several wars against the Rum-Seljuks, as the second and the third French-Seljuk War. These wars expanded France's influence on the Balcans and along the route to India. He also managed to conquer Jerusalem back, being the first European ruler since the Crusades to do so.
But he also made several wars in Europe: He interfered in the second Swiss-Savoy War, making both participants members of the Alliance des Alpes and thus his satellites. 1630-35, France and Denmark-Braunschweig fought the Netherlands in the anti-Dutch War. In the peace of Bremen, the Dutch gave Denmark-Braunschweig Nieuw-Nederland, and France got a good part of Flanders and all of Hennegau / Hainaut. 1660-64, he made the Luxemburgian War of Succession: France and its allies Venice, Bavaria, Switzerland and Nassau made claims for Luxemburgian territories, although they were highly doubtful. But with the absolutist François IV as French king, things like this didn't matter. Although Luxemburg had (most of) the HRE on its side, they were defeated and dismembered. Afterwards, he himself was elected Roman king in 1664.
1637, Poland had joined the HREGN with his help, and even got one electoral vote. Its German neighbors secretly claimed that this was for keeping them in check. Having his relative Charles / Károly IV elected Karl VIII, he pushed through the Reichsdeputationshauptbeschluss in 1653: All the free cities and other smaller territories of the HREGN were mediatized (given to bigger states). France itself received some lands too - the Grafschaften (counties) of Mömpelgard, Salm, Saarwerden-Saarbrücken, Veldenz and Sponheim, now extending into German-speaking territory.
Economically, he continued the policy of his predecessors to encourage trade with India. 1612: French merchants from Suez arrived in Ethiopia, started diplomatic connections. France took Djibouti to secure the way to India. 1620, the first French ship (coming from Suez) arrived in India. In the 1630s, French traders were allowed to trade in the Portuguese colonies in Africa, after Portugal had become independent with French help. 1664, French traders reached Canton in China.
The king also promoted the fine arts: In the castle of Fontainebleau, the first "Mexican Garden" was made in 1645. (Mexican gardens are influenced by the Aztec way of gardening / planting crops, putting many different colorful plants next to each other, forming geometrical patterns, with colors matching of course.) Under him, the style of Victorianisme reached its height.
But at the end, even his allies became aware of the rising power of France. When he annexed the German state of Baden in 1681, against all the rules, laws and traditions, even France's allies in Germany weren't willing to take these excesses anymore.
The anti-French War
The new king, François V, immediately tried to become Roman king like his father. 1682, the (several times postponed) election took place. (Several candidates, like crown prince Humphrey of England and king Humphrey I himself, had died at an inconvenient time, and the anti-French groups had to look for new candidates, since queen Kristina I of Sweden wasn't electable.) Finally, after a turbulent election, king Francisco II of Sicily was elected emperor Franz IV by Brandenburg, Prussia, Saxony, Jülich-Berg, the Netherlands, and the Palatinate (until then controlled by France), while François V ("the other Franz", as the Germans called him) was elected by Poland, Württemberg-Austria, Nassau, Braunschweig and Luxemburg. François V didn't want to accept the result of the election, which was admittedly quite unclear (are the votes of French-occupied Luxemburg and collegially governed Brandenburg valid?).
As a result, the anti-French War broke out. The king himself didn't witness the end of it anymore, although at the time of his death (1692) the French defeat was forseeable.
His successor François VI went on to fight a bit pro forma, but after he had a chieved a symbolic victory by breaking the siege of Bratislava, he was willing to make peace. In the peace of Amsterdam, he had to give up his allies Scotland, Mexico and Portugal.
Rebuilding French power
During 1717-23, France used the opportunity that Italy and Russia had defeated the Seljuks, allied with Persia and stroke against the Seljuks in the fourth French-Seljuk War. The French again occupied the Sinai and Aden. This time, Jerusalem was left alone.
1725, France and Persia made an agreement about Ethiopia. Both nations were allowed to trade with the land, it was not forced to switch religion in any way, but had to pay tribute to both of them.
France also expanded in India, taking Sri Lanka in 1728. And 1731, France founded the Indochinese League with Bengal, Pegu, Thailand, Khmer, Champa and Melaka. In 1738 however, the Swedes took Sri Lanka, after defeating a stronger French fleet surprisingly.
After the death of king (and Roman emperor) Eduard IV in 1722, the Netherlands were united with Sweden-Norway-Mecklenburg in personal union under his grandson. France objected the unification. 1724, it found an ally in Luxemburg; the Dutch War of Succession (also called Dutch War of Unification) began. It was resolved in 1729; France took Hainaut and parts of Flanders (again...), Luxemburg got Namur.
Then in 1735, Ludwig XVIII, last Wittelsbach duke of the Rhine Palatinate, died without heir. It wasn't completely clear whether France or Bavaria should get it. New emperor Heinrich IX of Brandenburg let the French unmisunderstandably know that he wouldn't tolerate them taking it. As a result, during 1736-42, the Palatinate War of Succession took place. Although the German and Swedish-Dutch armies couldn't defeat the French armies, when Spain and Britain threatened France, peace was made. The Palatinate went neither to France nor Bavaria, but to a grandnephew of the Bourbon king of Hungary. In case he and his brother should die without heir, the Palatinate would fall back to Bavaria.
After the council of Cork, France tried to get more influence on the Catholic church again. 1738, king François VI managed to smuggle pope Innocent from Rome to Avignon. The Italian king protested, but couldn't change it.