Il Campo in Siena ( which makes the list ) is one of my favourite squares, though it isn't "square" but shell shaped, sloping, and symbolically divided into sections. Ambrogio Lorenzetti's "Good government / bad government" murals are in the Palazzo Pubblico on one side, and the Duomo which has a gorgeous ornate interior with bold striped columns is on the other. You emerge from the narrow little streets into this vast and rather surreal open space ... what a feeling. The Sienese hold their colourful Palio there every summer, much as we have our delightful winter Slush Festival.
Il Campo is treeless too, thank goodness, just like the Piazza Navonna in Rome, another of Babel's favourite Italian promenades and people-watching places. It is home to a wacky Borromini church too.
There's also a circular "square" in Lucca, surrounded by 4 or 5 storey buildings, residential I think, that has a wonderful enclosed feeling to it. I think it was built on the site of an old Roman ampetheatre, though it is much more recent.