Status: town and former capital of the Hadramawt Kingdom
Age in years: 2,000ish
Number of medieval skyscrapers: about 500
Shibam is a walled city in the Yemeni desert that is famous for its mud brick high-rises. There are hundreds of such towers that are five to eleven stories high which were among the tallest in the history of the world when they were first built in the 16th Century. The architecture, which was orinigally developed as a defense against Bedouin attacks, has had the unintended consequence of making the town look really, really cool.
Of the town's architecture, Wikipedia says:
Shibam is often called "the oldest skyscraper-city in the world" or "Manhattan of the desert", and is the earliest example of urban planning based on the principle of vertical construction. The city has the tallest mud buildings in the world, with some of them being over 100 feet (over 30 meters) high, thus being the first high-rise (which need to be at least 75 feet or 23 meters) apartment buildings and tower blocks (except perhaps for the insulae of ancient Rome). The tallest building in the city is the mudbrick minaret which stands at 175 feet (over 53 metres) tall.
The city itself actually dates back to the 100s CE or so. It was intermittently the capital of the Hadramawt, a loose confederation of sultanates and emirates that extended (and extends today in the form of the Hadramawt Governate) from the southeastern coastal plain of Yemen into the Jol, the broad 4,500 ft. plateau in the Yemeni interior, and slopes down into the essentially uninhabited and liminally vague Empty Quarter. The climate is deserty, with occasionaly strong storms in April and September. Here's a link to the Shibam Urban Development Project, which exists to preserve the cultural heritage and promote sustainable economic development of the place, and has more information about the city.
I believe I would like to visit this place some day.