
(left) Banner of the Principality of Dombes; (right) Coat of Arms of the principality. Note the defining feature, the red diagonal stripe on the three fleurs-de-lys of the Bourbons. Both were used until the Principality's incorporation into France proper in the mid-18th century.
History
The territory of Dombes was part of Charlemagne's Frankish Empire until 843 AD, when the Treaty of Verdun assigned it to West Francia. However, the distance between Paris and the region meant that the region was almost completely autonomous from the Frankish kings, ruled only as a Frankish dominion called the Kingdom of Arles. When this Kingdom began to break up in the 12th century, the lords of neighbouring Bâgé took control of the region, passing it onto the lords of Beaujeu in 1218, who shared it with the lords of Thoire and Villars. All of these lords were loyal to the Kingdom of France, but ruled as de facto independent sovereigns.
The lords of Thoire and Villars, who controlled the southern part of Dombes, from the mid 14th century onwards, began moving up the chain of French succession; they were loyal to France during the Hundred Years War (1337-1453) and curried favours with the court in Paris, asserting their independence from the Holy Roman Empire. Meanwhile the lords of Beaujeu, who controlled the northern part of Dombes, came under the protection of, and later sold their land to the Duke Louis II of Bourbon in 1400, who also purchased the southern part two years later in 1402, uniting the two into a new feudal entity called the Principality of Dombes, the subject of this WCC.

(above) Map of the Principality of Dombes, whose borders remained more or less the same throughout its 350+ year long history; you can see the two, disconnected, "northern and southern" sections that were united in 1402 by the Duke of Bourbon.
Over a century later, in 1523, King François I of France accused the Duke of Bourbon of several felonies, after which his lands and titles (including the principality of Dombes and the title prince of Dombes) were confiscated by the King and ruled directly by the successive kings of France until 1561, when King Henri II returned Dombes to the Dukes of Bourbon. Those Dukes of Bourbon-Montpensier (as they were now called), would also rule the principality as its princes under the King of France for the next century, until in 1682 Duchess Anne-Marie Louise, the princess of Dombes, surrendered it to King Louis XIV in return for the release of her lover (yes, really!).
Louis XIV did not incorporate it into the Kingdom of France directly, but instead let his illegitimate son the Duke of Maine (no relation to the modern US state) have the title prince of Dombes, and rule it for him. The Duke of Maine would do so until his death in 1755, when his brother the Count of Eu succeeded him. The Count would later, in 1762, exchange the principality and the title of prince of Dombes for some private land with King Louis XV, who formally and fully incorporated the Principality into the kingdom of France permanently.
Coinage
Throughout the existence of the Principality of Dombes, the livre, with it's subdivisions of 20 sous to a livre, and 12 deniers to a sous, was used as its currency, as it was in France proper. Coins of the livre, created under Charlemagne had been used in the territory since 781 AD, long before the principality's 1402 creation. However, the Principality of Dombes, like some other Feudal issues, were permitted to strike their own coins, and they issued coins not with the busts of French monarchs, but with those of the reigning prince(ss) of Dombes. Apart from this and the slightly modified arms on the reverse, Dombes coins were identical to those issued in the rest of France, and their coins were usually legal tender and accepted throughout the Kingdom of France as well as in Dombes.

(left, right) Examples of deniers from Dombes; the left is from 1624, and you can see the small diagonal stripe (a baton) between the fleurs de lys; on the right 1650 one, you can see the "label argent" above the fleurs de lys, a heraldic motif on the Orléaniste branch of the Capetian dynasty.
https://en.numista.com/catalogue/dombes_principality-1.html