There's no way that I could choose the greatest, really. For one thing, it depends on the field. As for great Italians, there would be dozens on my list, many of whom have already been posted.
I would add some to the above lists, although I'm leaving many out. These are just people whose work has meant something to me personally and who weren't previously mentioned :
In painting, sculpture, and architecture: Rafaello, Giotto, Ghiberti, Donatello, Cellini, Bramante, Sansovino, Della Robbia, Verrocchio, Bronzino, Veronese, Guido Reni, Modigliani.
In literature and philosophy, of those not mentioned: Ovid, Livy, Horace, Thomas Aquinas, Pico della Mirandola, Macchiavelli, Benedetto Croce. Of "modern" writers: Pirandello, Ignacio Silone, Natalia Ginzburg, Elsa Morante, Alberto Moravia, Dario Fo, Primo Levi, Carlo Levi, Umberto Eco, Oriana Fallaci. I know he's already been mentioned, but Montale is very close to my heart.
In music, of those not mentioned: Monteverdi, Pergolesi, Mascagni, Paganini, Corelli, Riccardo Muti. Of modern performers: Caruso, Gigli,Tebaldi, Scotto, Galli-Curci, Corelli, Pavarotti. More "pop" performers: Mina, DeAndre, Lucio Dalla, Zucchero, Battisti, Battiato, Nannini.
In theater and film: Eleanora Duse, Rossellini, DeSica, Pasolini, Bertolucci, Tornatore, and yes, Sergio Leone. (Was Fellini mentioned?) As performers, there are so many, but among them, Anna Magnani, Sophia Loren, Marcello Mastroianni, Alberto Sordi,Toto.
Educators: Maria Montessori
Scientists: Luigi Cavalli-Sforza. There are many others, but like many scientists, they work more in obscurity.
Of the ancients not mentioned so far: Marcus Agrippa, if for nothing else, for the Pantheon, my favorite building in the whole world.
Saints: Caterina de Siena, patron saint of Italy, Roncalli, Papa Giovanni XXIII
Honorable mention: Anti-mafia crusading magistrates and others, like Giovanni Falcone and Paolo Borsellino