Just south of the Mercato Nuovo is a well-preserved pocket of medieval city. The Palagio di Parte Guelfa at Piazza di Parte Guelfa was the 13th-century headquarters of the Guelph party ('palagio' was the Florentine word for a building in between a tower house and a palazzo) and often the real seat of power in the city. They gleefully paid for the building with the profits from properties confiscated from the Ghibellines.
As Machiavelli wrote
This Parte Guelfa abused its powers exceedingly for party purposes, and the Parte Guelfa Capitani became more feared than the Signoria itself, and behaved most insolently towards it.'
So by the late 15th century it was dissolved. Before then, Brunelleschi added a hall on the top floor, now known as the Salone Brunelleschi, with a coffered ceiling by Giorgio Vasari. It also has a lunette by Luca della Robbia.
Image by Sailko, GNU Free Documentation License