Here are most of the cocktails in the IBA cocktail list ordered such that as you go down, at most one new cocktail ingredient each time is needed. (This doesn’t include other ingredients such as fruit juice.)

EDIT (2023): Brett Adams and Jacob Grier wrote a book with exactly this premise, turns out it was a good idea! Their order is: bourbon, gin, vermouth, orange liqueur, tequila, light rum, cognac, benedictine, campari, vodka, maraschino, rye, absinthe, chartreuse. Similar order, at least at the start, but vodka is much higher in my list and I am somehow missing rye.