I know that many wealthy and noble Medieval and Early Modern families such as the Orsini, Medici, and Fieschi produced multiple Bishops of Rome, but which family produced the most?
The Conti de Segni family produced four popes: Innocent III, Gregory IX, Alexander IV, and Innocent XIII.
The Medici family produced four popes: Leo X, Clement VII, Pius IV, and Leo XI.
Another (in)famous Roman family, the Borgia, had 3 popes (Calixtus III, Alexander VI, and Innocent X). The Orsini family produced Celestine III, Nicholas III, and Benedict XIII. Other "papal" families such as Barberini, Caetani, della Rovere, Borghese, Fieschi, Aldobrandini, Farnese, and Pamphilj had many cardinals but 0-2 popes each.
Source: George L. Williams, Papal Genealogy: The Families and Descendants of the Popes