The Khazar Khaganate
When speaking of Jewish history, the popular consensus is that Jews lived exclusively in diaspora since antiquity, with the Jewish Revolts of the Roman Period being the last period of Jewish political rule, and a short-lived one at that. Though this consensus is reductionist in more ways than one—there were multiple groups in that converted to Judaism in pre-Islamic Arabia, some of which became regional powers on the peninsula, and a few small kingdoms in Ethiopia—the Khazar Khaganate of the Early Middle Ages stands out to me as possibly the most powerful of these forgotten Jewish polities, and as the one I find the most interesting.
The Khazars were ethnically Turkic, and most likely originally lived in either the steppes of Central Asia or in the Ural or Caucasus Mountains. Very little is known about their early history, as is usually common with nomadic peoples.