Tier 1: The core four are absolutely required. Fighter, wizard, rogue, cleric. I could live w/o cleric, but it's D&D.
Tier 2: Paladin and some form of ranger come in very close, though my definition of ranger means it hasn't been "D&D" since 2e was published.
Tier 3: Psion, warlord, and witch. These aren't required, IMO, but I like the flavor and have had some variant that fits that role since the mid 1980s. In 1e, it was random psionics, the cavalier, and the witch class from Dragon. In 4e, it's psion, warlord (much better than the cavalier, but can pull the same flavor), and warlock. Editions between should be easy enough to trace.
Tier 4: Some concepts or class names have shown up repeatedly, but inconsistently. They often catch my eye but aren't always implemented well. Sometimes, they have precious little to do with a class by the same name in a prior edition. Regardless, I like some form of the concept, but could never call them "required". The big ones are shadowcasters, assassin, barbarian (1e hardy mage-foe != 3e berserker != 4e primal warrior), and duelist, but I'm sure I could come up with others, if pushed.
Bard is either tier 3 or tier 4. You can't say the 3e or 4e versions look much like the original, so it's inconsistent. But, the class has always been there, in some way. Usually in the PHB.