Pretty much all the highest % immigration countries are monarchy that I can think of, since in those country another tax payer is an easy win and immigrants that cause problem can be instantly booted so there is very little downside to taking anybody with $1 or a job who cares to come.
Top Countries by Percentage of Immigrants (approximate recent figures):
Qatar: Around 77% (or 76.7%).
United Arab Emirates (UAE): Around 74-88% (some sources show higher figures for earlier years).
Kuwait: Around 69-73%.
Bahrain: Around 55%.
Singapore not far behind (~40% from memory), a one party state but with voting, sometimes described as essentially an elected recallable monarchy. Also note most of those countries have relatively low emigration rates of native citizens.