Near the end of the movie where the wolf pack is returning to the minivan, there is a billboard with Eddie (the man who ran the wedding chapel in the first The Hangover (2009) movie) in the background.
In every film of the trilogy, including this one, there are pigs somewhere. In The Hangover (2009), as Phil, Alan and Stu wake up in the trashed hotel suite, there is an inflatable pig in the bubble bath. In The Hangover Part II (2011), during the car chase through Bangkok, Chow crashes into a pig hanging on a meat stand and Stu gets covered in some of it. In The Hangover Part III (2013), Marshall's thugs wear pig masks.
In the scene at the end of the film where the wolf pack get out of the limo and cross the street to return to Phil's minivan, the shot of them crossing the street clearly parodies the cover of the album by The Beatles, Abbey Road. After the movie's completion, a mural of this shot was painted onto the wall adjacent to the building.
There is an exotic animal appearing in each Hangover film, because of Alan. In the first film it's Mike Tyson's tiger. In the second it's the drug-dealing Crystal the Monkey, and in this one it's the giraffe, seen at the start of the film.
The wrestling match on the tv at the pawn shop was CM Punk vs John Cena on Raw from Feb. 25th 2013. The winner (Cena) would go on to face The Rock at that years Wrestlemania in New York. The logo for Wrestlemania can be seen in the background.