It does not.
See 1st PHB p. 32, The Multi-Classed Character, and p. 33, The Character With Two Classes.
Neither does it prohibit adding more types. The Thief, Paladin, Assassin, Monk, Druid, Ranger, Illusionist and Bard, and the half-elf, gnome and half-orc as player-characters, all came from...