Final answer:
The Persian Empire reached its greatest extent under Darius I, who expanded its territory and organized the administration with a strong centralized bureaucracy and regional governors known as satraps.
Step-by-step explanation:
The Persian Empire reached its largest extent under the rule of Darius I. He inherited the territories conquered by his predecessors and further expanded the empire's reach. In 522 BCE, following Cambyses' death, Darius I became king and took on the monumental task of organizing and consolidating the vast empire. He incorporated new territories, such as the northern part of the Indus river valley and Thrace, making the empire the most extensive it had ever been. Darius I's conquests and administrative reorganization allowed the Persian Empire to flourish from the Balkans to the Indus River valley, creating an imperial state with a strong centralized bureaucracy and regional satraps.