One of the “Slim, black” versions of the North American (NTSC) PS2 console.  [Note:  The model SCPH-79001 CB has an EXTERNAL power supply. By contrast, the more recent model SCPH-90001 CB has an INTERNAL power supply.]
The “Ceramic White” (special-edition model SCPH-79001 CW) PS2 Slim console, which I use to play my PS2 (and PS1) games. [Note:  My brand-new specimen didn’t come with the pictured “stand” that supports the console in the pictured, optional “vertical” position; hence I separately ordered that stand  from a different online vendor.]
Pros: The library of collectible, fun-to-play PS2-compatible “racing” games on disc is huge. And many titles are priced low.
Cons: Most of the extant PS3 consoles aren’t backward-compatible with PS2 games.  And with virtually all PS2 consoles, in order to “save” games, you’ll need to have at least one reliable “PS2 memory card”– preferably an original Sony-brand [or perhaps Nyko] card, because cheap “generic” cards are notoriously unreliable.
 
Preface  (Feel free to skip ahead to the lists of games.)
According to Wikipedia, the PlayStation 2 is

“… the best-selling video game console in history…. More than 3,874 game titles have been released for the PS2…, and more than 1.5 billion copies have been sold.”
Even factoring that the above-cited numbers of titles and sales pertain to the entire world (not just North America), it’s obvious that – 15 years after the PS2’s advent – American buyers of “racing and driving” games should have no trouble finding many affordable titles for their PS2 collections.
Even if no brick-and-mortar stores in your neighborhood still stock PS2 games, it’s generally easy to order such games online via Amazon, eBay, Half.com and other sites.

I didn’t own a PS2 console until January 2010. And I didn’t start collecting PS2 racing/driving games till 2013; but by now I’ve bought 124 such PS2 titles [plus 121 such titles in PS1 format, which is compatible with PS2 consoles]; and I would’ve bought still more PS2 racing games if I hadn’t already collected certain “cross-platform” titles in their GameCube, Wii, Xbox and/or Xbox 360 editions.
During the months when I was painstakingly building my American PS2 games collection, I felt continually frustrated by the most widely disseminated, incomplete lists of “racing” titles for this or that particular system. Thus I hope my below — presumably comprehensive — list of recommendable PS2 racing/driving-game titles may be of help to some fellow collectors of this genre [though it’s possible I’ve overlooked a title or two that merit inclusion].
However, note that I have three key criteria determining (or at least influencing) which PS2-compatible racing games ended up on the list (and in my personal collection).

First, titles like “Driver” or “Grand Theft Auto” [which I myself neither play nor collect] are primarily role-playing, not racing, games. Thus I generally don’t include such titles on this list.
Second, excessively puerile/adolescent/cartoony graphics, audio or themes normally don’t appeal to this adult gamer; so, you generally won’t find such titles on this list.
Third, if a game simply incorporates hopelessly clumsy mechanics – or glaringly ineptly rendered audio/graphics – that keep me from enjoying the overall experience, that title is anathema. [Here too, however, there’s a gray area, insofar as my below list does encompass a small minority of “borderline-acceptable” titles that somehow manage to amuse me enough to merit inclusion.
As I’m American, my below list predictably comprises only NTSC editions; some games may bear different titles (or perhaps weren’t released) in other world regions. [Consult Wikipedia for details.]
Finally, my intention isn’t to discuss every game below; but the simple fact that these are the titles that made my list means each somehow merits a passing grade (though in several instances just a “D”). So, use the following list judiciously,  perhaps consulting pertinent reviews, articles or videos at the worthy Amazon, eBay, IGN, Wikipedia and YouTube to decide for yourself if a certain title belongs in your collection.
RECOMMENDABLE RETAIL PS2 “RACING” GAMES RELEASED ON DISC IN NORTH AMERICA:
[Except where noted below, I own a PS2-edition specimen of each of the following titles.]
(To view a picture of a particular game at Amazon, click its title.)

1. [Note: I opted instead for the (Wii-compatible) GameCube edition of this game.]
10.   [Note: I opted instead for the (Xbox 360-compatible) original-Xbox edition of this game.]
    
11.      [Note: Be aware that the “car handling/steering” in this game (not only in this PS2 but also the virtually identical Xbox and GameCube editions, which I’ve likewise collected) is notoriously touchy/difficult! “Casual/occasional” gamers would do well to skip this title.]
12. [Note: I opted instead for the (Xbox 360-compatible) original-Xbox edition of this game.]
13. [Note: I opted instead for the (Xbox 360-compatible) original-Xbox edition of this game.]
14. [Note: I opted instead for the (Xbox 360-compatible) original-Xbox edition of this game.]
16.    [Note: I opted instead for the graphically superior Xbox 360 edition of this game.]
    
31. [Note: I opted instead for the (Xbox 360-compatible) original-Xbox edition of this game.]
32. [Note: I opted instead for the (Wii-compatible) GameCube edition of this game.]
35. (2008)
36. (2002)
37. [Note: I opted instead for the (Xbox 360-compatible) original-Xbox edition of this game.]
38. [Note: I opted instead for the improved 2008 re-release of this game for Xbox 360, titled .]
45. [Note: I opted instead for the (Wii-compatible) GameCube edition of this game.]
53. [Note: I opted instead for the graphically superior Xbox 360 edition of this game.]
54. [Note: I opted instead for the (Xbox 360-compatible) original-Xbox edition of this game.]
55. [Note: I opted instead for the (Wii-compatible) GameCube edition of this game.]
64. [Note: I opted instead for the graphically superior Xbox 360 edition of this game.]
74. [Note: Don’t confuse this PS2 game with the similarly titled original-Xbox release  (which is a distinct– and, by the way, not “360-compatible” — game that I play solely via my original-Xbox console).]
77.   [Note: Unlike the somewhat similar PS2 game , this game doesn’t offer competitive bicycle racing. But its “stopwatch” mode does offer some seemingly more realistic against-the-clock solo racing, which real-world mountain-bike riders would probably appreciate more than would most casual gamers.]
78. [Note: I opted instead for the (Xbox 360-compatible) original-Xbox edition of this game.]
81. [Note: I opted instead for the (Wii-compatible) GameCube edition of this game.]
82. [Note: I opted instead for the (Xbox 360-compatible) original-Xbox edition of this game.]
83. [Note: I opted instead for the (Xbox 360-compatible) original-Xbox edition of this game.]
84. [Note: I opted instead for the Wii and Xbox 360 editions of this game.]
85. [Note: I opted instead for the (Xbox 360-compatible) original-Xbox edition of this game.]
92. [Note: I own not only this excellent PS2 edition but also the very similar Wii-compatible GameCube.]
97. [Note: I own not only this PS2 edition but also the GameCube, whose gameplay content differs somewhat.]
98. [Note: I opted instead for the (Wii-compatible) GameCube edition of this game.]
99. [Note: I opted instead for the (Xbox 360-compatible) original-Xbox edition of this game.]
100. [Note: I opted instead for the Xbox 360 edition of this 2005 game.]
101. [Note: I opted instead for the Xbox 360 edition of this game.]
102. [Note: I opted instead for the Xbox 360 edition of this game.]
103. [Note: I opted instead for the Xbox 360 edition of this game.]
104.   [This is a 2005 release. Also see (directly below) its superior 2007 sequel.]
106. [I prefer the bike control (“steering”) in this PS2 edition to that of the Wii edition (which I likewise own).]
108. [Note: I opted instead for the (Wii-compatible) GameCube edition of this game.]
111. [Note: I opted instead for the (Wii-compatible) GameCube edition of this game.]
118. [Note: I opted instead for the Xbox 360 edition of this game.]
119. [Note: I opted instead for the Xbox 360 edition of this game.]
126. [Note: According to IGN, (for GameCube) “is basically just ‘Hostile Territory’ with some additions and improvements.” Hence I opted to skip this PS2 release and get the enhanced GameCube version.]
127. [Note: I opted instead for the (Xbox 360-compatible) original-Xbox edition of this game.]
134. [Note: I opted instead for the (Xbox 360-compatible) original-Xbox edition of this game.]
135. [Note: I opted instead for the (Wii-compatible) GameCube edition of this game.]
     
137. [Note: I opted instead for the (Xbox 360-compatible) original-Xbox edition of this game.]
140. [Note: When played in its “career/season” mode, the PS2 edition of this little low-budget game really entertains me– as does the Wii edition, which utilizes the Wiimote’s “motion-control steering” surprisingly effectively.]
145. [Note: I opted instead for the (Xbox 360-compatible) original-Xbox edition of this game.]
146. [Note: I opted instead for the (Xbox 360-compatible) original-Xbox edition of this game.]
148. [Note: I opted instead for the Xbox 360 edition of this game.]
165. [Note: I opted instead for the (Wii-compatible) GameCube edition of this game.]
166. [Note: I opted instead for the (Wii-compatible) GameCube edition of this game.]

Note: The following PS2 title is somewhat more a “combat-driving” than a “racing” game. [I opted instead for its (Wii-compatible) GameCube edition.]
Also note: The following two PS2 titles [both of which I opted to get in their “Xbox 360-compatible,” original-Xbox editions] are definitely more “combat-driving” than “racing” games, but they still might be of interest.
     
Also note that there are two “Stuntman” titles available for PS2, including the 2002 PS2-exclusive debut , and the 2007 sequel [which I opted to get in Xbox 360 format]. While I do admire certain things about those “Stuntman” titles, I’m reluctant to characterize them as “racing” games or to declare them highly recommendable for the majority of players. But I do suggest that you consult several YouTube videos/reviews to help you decide if the PS2 “Stuntman” series is for you.
(PS2-COMPATIBLE) PS1 RACING GAMES
Of course, in addition to all the above PS2-format game discs, a PS2 console can play essentially any “original PlayStation” (PS1) game disc. [Thus the total number of recommendable PS2-compatible North American (NTSC) racing and driving games is actually around 292!]
To see my complete list of such PS1 racing titles, click the following link:
The 121 Best RACING Games for PS1
 
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You may also enjoy the following:
The 99 Best RACING Games for Original Xbox
The 48 Best RACING Games for GameCube
The 81 Best RACING Games for Nintendo Wii
The 135 Best RACING Games for Xbox 360