Simulation (or close enough to be simulation):
1. Dirt Rally - With a wheel, this is magnificent. It's so much fun and the driving is intense. I find it hard to play for more than an hour or so at a time because it takes so much concentration. Once you lose focus, the race is just about over because its so hard to make up the lost time and you end up gambling a lot more which leads to more mistakes. The only thing I would have liked to see would be a more fully fleshed out career mode, as right now its literally one results screen after each race.
2. Gran Turismo 3 - This game has long been surpassed in terms of simulation racing, but it was probably the peak of that 'type' of racing game which covered a variety of motorcars. And it looked damn good at the time.
3. Forza Motorsport 3 - This game feels better to play than GT3, GT4 (and possibly GT5), but it was more of the same type of thing. It's also the last Forza I got, but it was fun for the 20 hours or so I put into the game.
Honourable Mention - Assetto Corsa - Probably the most simulation like driving I've played in a game, but the career mode is really sterile, hence I haven't really put much time into the game. But the driving really is outstanding. I haven't played rFactor, Raceroom or iRacing, but Assetto Corsa driving model beats everything else out in the market.
Arcade : (or close enough to be arcade)
1. Project Gotham Racing 2 - This game had EVERYTHING. Amazing tracks, decent career mode, online. Cities that haven't been seen in other racing games since or to that level of detail. Barcelona, Stockholm, Washington, Chicago, Hong Kong, Sydney, Edinburgh, Yokohama. Brand new cities that were not part of PGR1. I can't wait to go back home and pull this game out again. This and Halo MADE the Xbox worth getting. My favourite racing game of all time.
2. Outrun 2 - Going to put this ahead of RR6, simply because the driving mechanics are superb. It's just crazy fun to rip out drifts. It's a short game, but the career mode is pretty fun and a challenge.
3. Ridge Racer 6 - Couldn't decide between this or RR7, but went with this as RR7 was more of the same. I had a race, possibly the best race I've ever had in a video game, where I was neck and neck with the AI for over two laps, before eeking out a victory in the last third of the race. Still have the replay for it I think. Incredible game, with incredible boost mechanics that made you learn the tracks and where to boost.
Honourable mention - Sonic Transformed I was shocked how amazing this game was for a kart racer. Even after playing MK8, Sonic Transformed is easily the better game. Great tracks, fun transformable vehicles. It's probably too hard for some though.
Dishonourable mention - Burnout 3 Seriously bewilders me, how many mentions this gets as people's favourite racer. This game has a great first 5 hours, then the entire thing falls apart when you get good, broken AI cars, races where you boost the entire way with the AI always 4 seconds behind ready to pounce. I allege that this game doesn't even have racing, and racing only happens if cars are actually within a certain distance from the player car. One day i'll get an Elgato, and record gameplay from this to prove it, but as such I can't be anymore critical than this without proof. There are some posts online that agree with me, but there are no youtube videos of it.
I'm also looking forward to playing Forza Horizon 3 when it gets a little cheaper. It looks amazing.