Rabbi Dr. Yosef Burg z’tz’l was a big Talmid Chacham. He founded the National Religious Party and served in the Cabinet and the Knesset for years.
Rabbi Yehuda Amital z’tz’l was a big Talmid Chacham who served briefly in the Cabinet.
Some of the past Chief Rabbis of Israel have been among the gedolim of the generation. Rabbi Ovadia Yosef z’tz’l was one of the gedolim of all generations; Rabbi Yitzhak Herzog z’tz’l was Chief Rabbi of two different countries.
There is a lot of hakachic literature about Karov HaMelech. There are things that people close to the rulers must do that would not be permitted for the rest of us.