Grease. Specifically (according to Pliny), goose grease or poultry grease, or wool grease (lanolin) mixed with nut galls. (Pliny also recommends rubbing your lips with spiderwebs, but I know what I'm trying first.)
Other animal greases and fats (lard, buffalo, bear, possum, whatever you have to hand) were used liberally, as were olive oil and other vegetable oils. Beeswax was used for a very long time. Earwax was used for a much longer time before that and came in handy in areas where beeswax was not readily available.
Glycerin joined the arsenal of skincare products in the late 18th century, and Vaseline was on the market several years before the first lip balm tubes came out in the 1880s.
I remember reading in a Simon Schama's An Embarrassment of Riches back in undergrad that people would put clove oils on their chapped hands in the winters, but if you couldn't afford clove oil, a mix of lanolin and wax was used instead.
hi! /u/intangible_tangerine listed a few more solutions here