He was the first European to see the holy cities of Mecca and Medina and live to tell the tale, something he accomplished by disguising himself as and adopting the language and customs of a wandering Dervish -- a daring feat which risked certain death if he was discovered. He managed to repeat the feat with various other forbidden cities around the world, though not without injury.
He produced, to my mind, by far the most entertaining (though apparently far from accurate) translation of the Thousand and One Nights, and the erotic classic, the Kama Sutra, along with various other works.
"Throughout his life, Burton continually sought passionately for "Gnosis," which he pursued around the world in myriad forms. He was a student of the Kaballah and of Hermeticism. He was an initiated Nagar Brahmin and Kadiri Dervish; he became a member of the Ismaili sect, which claimed decent from the Assassins; and he was at various times a formal convert to Hinduism, Tantrism, Roman Catholicism, Sikhism and Islam."[2]
He was a founder of The Cannibal Club[3], where men (of course they were all men) could openly discuss "subjects deemed deviant by society".
He was also a through and unapologetic racist and colonialist, almost to the point of caricature, which makes his works (which typically contain many personal notes and digressions) hard to swallow for most modern readers.
After his death, his wife burned much of his unpublished work and diaries (presumably because of their frank sexual content), so we may never see his most scandalous work.
[1] - https://en.wikipedia.org/wiki/Richard_Francis_Burton