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What sort of driving style you looking for?
Assetto Corsa is awesome but definitely better with a wheel (I played it for quite a while with just a controller though). GT Legends and Race07 are also both great sim-like games which still hold up to this day.
On the complete arcade side, I love the Trackmania games, Trackmania United Forever is a classic and is half price at the moment (it's comprised of 7 different environments, 1 of which is available as a free standalone 'Trackmania Nations Forever'), there's also a newer bunch which are basically the same environments with richer graphics and slightly updated handling models.
Driver San Fransisco is a damn good driving game, excellent soundtrack too (I realise that's not high praise coming from me, but trust me it's good).
MX vs. ATV Reflex is a good off-road racer with fancy realistic mud physics.
I've recommended it quite recently but Sonic & All Stars Racing Transformed is a great kart racer.
Goes without saying that Crazy Taxi is worth picking up, and along the same lines, OutRun 2006 is one of my favourite games but is not for sale digitally anymore due to Ferrari licence expiration.
Absolute Drift is a nice relaxing driving game with a sort of 3D/isometric view, the developer is currently working on a very cool Group B era rally game in the same vein. There's a demo available here: https://funselektor.itch.io/art-of-rally
Recommendation for a driving game? I'm about 20 wins short of the Elite license in Paradise City and not sure I CBA with the stunt runs to get them all done. What PC/Controller/On Steam games are similar that people can recommend?