Final answer:
The Arab-Islamic Empire achieved its pinnacle of fame and glory under Caliph Haroun al-Rashid, whose reign was marked by vast expansion and cultural exchanges with other empires such as that of Charlemagne.
Step-by-step explanation:
The Arab-Islamic Empire reached the climax of its renown and brilliance under the rule of Haroun al-Rashid, during the time period from 786 to 809. The empire not only expanded in terms of land, covering areas larger than the Roman Empire, but also in terms of cultural and diplomatic influence, evident through interactions with figures such as Charlemagne, the European king. Haroun al-Rashid's rule signified a time of great power and cultural exchange, seen through the exchange of gifts between empires despite underlying political motives.