I know Paris better than Rome, and it is similar, just not as many layers deep. From the castle walls under the Louvre [1] down to the curved walls in downstairs bathrooms (a little restaurant near Notre Dame, likely just an old cellar, and probably only 400 years old, not 2000). And before you dismiss the history as happening only in Europe and only centuries ago, San Francisco once moved its cemeteries along the old main road out of town (Colma along El Camino Real) and Seattle already has a second layer [2]. History repeats itself, or rather humans are the same, with the same habits and patterns (like that other thread about a one star review on a Sumerian clay tablet [3]).
Another interesting story about Rome: because of all the layers of history, and the richness of Ancient Rome, artifacts have been unearthed for years. Imagine living in the 1500's or 1600's and hearing about a great statue that was unearthed, and much finer than most anything existing at the time. This was the case of Laocoön and his Sons (1506) [4] and the Dying Gaul (early 1600's) [5]--their rediscovery is part of our history. Of course, Roman statues are still unearthed all over the old Roman Empire to this day. In an odd twist, so much time has passed that old collections of Roman art have been lost again and then recovered recently [6].
[1] https://en.wikipedia.org/wiki/Louvre_castle#The_remnants_of_...
[2] https://en.m.wikipedia.org/wiki/Seattle_Underground
[3] https://news.ycombinator.com/item?id=15669759
[4] https://en.wikipedia.org/wiki/Laocoön_and_His_Sons
[5] https://en.wikipedia.org/wiki/Dying_Gaul
[6] http://www.telegraph.co.uk/news/worldnews/europe/italy/11285...