There are currently no monsters with more than one creature type
The Monster Manual (page 7) includes a list of all creature types (cutting out the descriptions):
Aberrations, Beasts, Celestials, Constructs, Dragons, Elementals, Fey, Fiends, Giants, Humanoids, Monstrosities, Oozes, Plants, and Undead
After a rather large amount of scrolling through the results pages on dndbeyond I have come to the conclusion that there do not exist any monsters with multiple creature types.
There are creatures, like the Quasit, that are shapechangers, or are a specific subset of a type (in this case a Quasit is a fiend, but more specifically, a demon); however, shapechanger is not a creature type in and of itself, nor is demon. Similarly a Swarm of Bats has the creature type Beast, it being a swarm doesn't change that.
There are also creatures like the Tanarukk, a fiend (demon) and an Orc, except Orc is not a creature type, Humanoid is, but the Tanarukk are not humanoids. The description of this creature includes the following:
[...] The demon lord Baphomet gladly shares the secret of creating tanarukks with those who entreat him for power. The process corrupts an unborn orc of the tribe, transforming it at birth into a creature much more savage than an orc [...]
Despite there not existing any monsters with multiple creature types, there are unpublished races with multiple creature types. For example, the Unearthed Arcana Centaurs and Minotaurs (pdf direct link) include the following feature:
Hybrid Nature. You have two creature types: humanoid and monstrosity. You can be affected by a game effect if it works on either of your creature types.
Notably the later officially published versions of these races do not have multiple creature types. So outside of this Unearthed Arcana I do not know of any creatures (NPC or PC) that have multiple creature types. I also do not know any way of adding a creature type onto an already existing creature without removing the one they currently have.