This answer is preliminary, incomplete and therefore quite possibly wrong, and it is only based on the movies, tv shows and video games. For a very complete list of Batmobiles, see this interesting and excellent resource.
To try and answer your question, here is what I could count when you take into account the continuities:
Continuity 1 - 1960's Continuity
1 type of Batmobile, the same one depicted in the 1966 movie and the 1966-1968 tv series
It was not depicted in the 1967 Batgirl tv short
An animated version of this Batmobile was depicted in the 1977 tv series The New Adventures of Batman
Continuity 2 - Batman 1990's movies tetralogy
- 3/4 types of Batmobile depicted, a different one seen in each of: Batman (1989), Batman Returns (1992), Batman Forever (1995), Batman & Robin (1997) - The Batmobile shown in Batman Returns appears identical to that in the 1989 film but shows new abilities (Batmissile mode), so it is unclear if it is a different model
Continuity 3 - Batman 2000's Nolan Trilogy
- 1-3 Batmobiles depicted depending on how different you think they are, as seen in Batman Begins (2005), The Dark Knight (2008) and The Dark Knight Rises (2002)
Continuity 4 - Batman the Animated Series
2 types of Batmobile depicted
1 in BTAS (1992-1995), but none depicted in either Mask of the Phantasm (1993) or Subzero (1998) (only Batplanes were shown in these two animated movies)
1 type, different from BTAS, depicted in both the New Batman Adventures tv series (1997-1999) and the movie Mystery of the Batman (2003)
Continuity 5 - Batman Beyond
- 1 type of Batmobile depicted, a sort of flying/hovering batmobile shown in both the tv series (1999-2001) and the Return of the Joker movie (2000) - might be a Batplane instead
Continuity 6 - The Batman
- 1 type of Batmobile depicted, the same one shown in both the tv series (2004-2008) and the movie The Batman vs. Dracula (2005)
Continuity 7 - Batman the Brave and the Bold
- 1 type of Batmobile depicted in the 2008-2011 tv series
Continuity 8 - Warner Brothers Video Game Series
- 4 types of Batmobile depicted, only one of them playable in the 2015 game Arkham Knight; the others are Arkham Asylum (2009), Arkham City (2011) and Arkham Origins (2013)
Continuities are not listed in any particular order.
As mentioned, this list is far from exhaustive, there are lots of other continuities in the comic books and even in live media. I don't know how continuous the crossover movies are (the Justice League ones where Batman appears) or how coterminous they are with various continuities in the baseline media.
Therefore, from what I can see, and in anticipation of a better/more complete answer, it seems to be a tie between the 1990's tetralogy (depending on whether or not the 1992 film Batman Returns actually employed a different model from the 1989 film) and the Warner Brothers video game series, with 4.
Apart from my own media and knowledge, I also used these sources: 1, 2, 3 and 4.
Another resource can be found here.