If I had to pick a SINGLE favorite... Probably would be The Crew for now simply because of the Multiplayer aspect. Just being able to tackle missions and go on virtual road trips is a lot of fun.
But, here's a list of the racing games that I think are noteworthy and why, in no particular order:
Daytona USA (Sega) - Arcadey racing goodness, need I say more? The 360 port is amazing and worth getting.
Sega Rally Championship (Sega) - I like my Rally/WRC racing and SRC is the granddaddy.
Initial D Special Stage (Sega Rosso) - A Japan-only PS2 release, a very good port of Initial D Arcade Stage 3.
Wangan Midnight Maximum Tune 3 (Namco) - Out of every street racing game I've ever played, this has some of the
BEST game feel ever, plus the best camera handling/movement while doing a drift. If you ever see this in a arcade, play it.
Dead Heat (Namco) - The FNF ricing is a bit out of control in this one, but the game feel is really good. Does suffer from some rubber banding issues though. A good game nonetheless.
Distance (Refract Studios) - Driving on walls and stuff. It's on Steam. Go buy it.
Carmageddon (Stainless Software Ltd) - Take the movie Death Race 2000 and make it a game, and you have Carmageddon. Both the original and it's sequel, Carpocolypse Now, come highly recommended.
DiRT 3 (Codemasters) - Another excellent WRC game with lots of different cars and tracks. Unfortunately suffers from AWFUL Games For Windows Live DRM.
Auto Modellista (Capcom) - A PS2 game that acts and feels like Gran Turismo, but it's cel-shaded cars. Worth looking into if you like Simulator-arcade hybrid racing.
Auto Club Revolution (Eutechnyx) - PC exclusive that is a free-to-play game Simulative racing game in the same style as Gran Turismo. Does suffer slightly from Pay-to-win, but it's still in open beta at the moment. Worth checking out since it's free.
BeamNG.drive (BeamNG) - Mostly still a Tech demo for now, but it attempts to by a hyper-realistic soft-body physics engine that tries to accurately represent car damage.
Burnout 2 (Criterion Games) - Probably the best Burnout game in the series. It's really good.
Burnout Paradise (Criterion Games) - Open world Burnout with events? I like it!
Ford Racing 2 (Razorworks) - Yes, it's a bargain bin walking advertisment for Ford. It's not an amazing game per se, but it's fun in short bursts.
Gas Guzzlers Extreme (Gamepires) - Mario Kart and Mad Max had a baby and it has violent tendencies. Couch multiplayer is blast with this one, and the game is easy to pick up and play.
RUSH (Atari Games/Midway) - All three main RUSH games are excellent arcade goodness and you should play them ASAP.
Sonic & All Stars Racing Transformed (Sumo Digital) - It's like Mario Kart but has better item balance. A great kart-style racer with all those wacky items and what not.
Top Gear (Gremlin Graphics/Kemco) - SNES racers that are really fun to play, and the music is awesome in all three SNES games.
Top Gear Rally (BOSS Studios/Kemco) - Excellent Rally racing, and pretty impressive graphics for the N64 era. The sequel for N64 is awful though, avoid it.
F-Zero (Nintendo) - Hyper-fast hovercar future racing. All of the games are good except Maximum Velocity.
California Speed (Midway) - It's like Crusin' but with less rubber banding problems and better game feel, I personally think. The N64 port is worth checking out if you can get it for cheap.
...There's honestly probably more that I like but I honestly can't remember at the moment, haha.