Final answer:
The chronological order is as follows: the rise of the Neo-Assyrian Empire, the collapse of the Assyrian Empire and the foundation of the Neo-Babylonian Empire by Nabopolassar, the reign of Nebuchadnezzar II with his campaigns and construction projects in Babylon, and the eventual fall of the Babylonian kings to Cyrus II of Persia.
Step-by-step explanation:
To place the events in chronological order from earliest to latest, we will begin with the Assyrian Empire, which gained power in the early ninth century BCE under Ashurnasirpal II. By the end of Tiglath-Pileser III's reign in 727 BCE, the Neo-Assyrian Empire was the dominant power in the Near East until around 600 BCE, when the Neo-Babylonian Empire, led by the Chaldeans, and the Medes pressured it into collapse.
Nabopolassar founded the Chaldean or Neo-Babylonian Empire after the decimation of the Assyrian Empire and reigned as king from 626 to 605 BCE. His son, Nebuchadnezzar II, succeeded him and is most notable for his military campaigns and ambitious construction projects, including the possible construction of the Hanging Gardens. He also led an assault on Jerusalem. This series of events transpired during the late 7th century and into the 6th century BCE.
Finally, after the Babylonian kings Nabopolassar and Nebuchadnezzar II claimed much of the Assyrian territory, they rebuilt Babylon. The Neo-Babylonian Empire then ruled an extensive territory in the Near East until its defeat by Cyrus II of Persia in 539 BCE.