It depends on the degree of fineness you wish to split things with--that is, where you set the bar for what is "baseline."
With the furthest zoom, coarsest grain? There are only two classes, Warrior and Caster, and everything else is an intersection between them or a specialization of them. (E.g., Cleric is a supportive Caster, Rogue is a skillful Warrior, etc.)
With the "traditional" grain? Fighter, Magic-User, Cleric, Thief. You could already view Cleric as a F/MU hybrid with support-focused spells, but many will argue this is a more appropriate stopping point (pretty much only due to tradition, not due to anything inherent to the archetypes.) From there, Barbarian, Druid, Ranger, Paladin, Bard, Sorcerer, Warlock, and Monk are all composite to some extent, just differing as to which elements. Artificer is sort of in an awkward place here because it's sorta like a Wizard/Rogue blend, but sorta distinct as well.
With a "modern" grain? Fighter, Wizard, Cleric, Rogue, Druid, and maybe Monk. (Ironically, you can actually argue that the Barbarian as it exists today borders on being a Fighter/Monk hybrid, since it's picked up the Monk "high defense without armor" feature, even though originally Barbarian was clearly a Fighter/Thief with some arguable Druid elements thrown in for fun.)
I think the inarguable "composite" classes are: Barbarian (Fighter/Rogue or Fighter/Druid), Bard (Fighter/Rogue/Wizard/Cleric), Ranger (Fighter/Druid or Fighter/Cleric), Paladin (Fighter/Cleric), Warlock (Cleric/Wizard, or Fighter/Cleric/Wizard for blade warlocks specifically.) If the playtest Sorcerer had survived, it would've slotted in here perfectly, being a Wizard mixed with some other class (Fighter for Dragon Sorcerer, likely Rogue for Chaos and/or Shadow, Cleric for Divine Soul, Druid could be a plausible option for Storm, etc.) As it stands, Sorcerer is mechanically not much different from a Wizard other than metamagic, and metamagic doesn't feel uniquely Sorcerous.
The arguable "composite" classes are Monk (Fighter/Rogue) and Druid (Cleric/Fighter?), with the latter having a more clearly distinct niche, as recognized by the Primal power source back in 4e.