The PHB mentions Improvised Weapons, which do not necessarily belong to either category:
Sometimes characters don't have their weapons and have to attack with whatever is at hand. An improvised weapon includes any object you can wield in one or two hands, such as broken glass, a table leg, a frying pan, a wagon wheel, or a dead goblin.
Often, an improvised weapon is similar to an actual weapon and can be treated as such. For example, a table leg is akin to a club. At the DM's option, a character proficient with a weapon can use a similar object as if it were that weapon and use his or her proficiency bonus.
Natural Weapons (such as the Lizardfolk's jaws) are also a thing, and according to Jeremy Crawford they too are neither martial nor simple.
As enkryptor mentioned, some monsters use weapons that are not explicitly categorized.