I still fire up the 360 arcade remake OutRun Online Arcade every once in a while. It has a charm that really appeals to my boyhood dreams and I don't care if the game offends anyone. Unfortunately I think Ferrari pulled the license after it lapsed so it isn't for sale anymore.
Arcade racing games
I well try to mention some games that have not already been listed. Thanks to the other posters for highlighting some games I didn't know.
I love Test Drive Unlimited because you could buy cars and garages on a life size (I guess) replica of Hawaii. It was a ground breaking open world driving game (before Burnout Paradise I might add). The sequel Test Drive Unlimited 2 was slightly disappointing because the first one was so great but I enjoyed it anyways (truly unique exotic cars that you can test drive, buy and customize). Unfortunately it probably didn't sell well because the studio closed.
Forza Horizon has a great 'campaign' with unique events. The tuning from Forza Motorsport is removed but you can still customize the components.
Slightly Mad Studios make car collecting games (that I could never grasp the handling of) like: