Recomend a Racing Game

big rigs over the road racing

Road Rash.
The Emperor's New Groove (Kuzco vs. Kronk)

Mario Kart 8 :P

Street Racing Syndicate

The Crew has been a pretty good filler for my racing fix for the past few weeks. Worth a look.

I like Grid 2

Old Need for Speed? :P
I'd consider those new Need for Speed, the old stuff was something like original Hot Pursuit and the like. The series started getting worse with those titles you listed. :(

My own suggestions are a bit older (one was a DOS release) with two on GOG, one physical disc only. All pure arcade racing.
Outrun 2006: Coast 2 Coast (physical only) - gameplay
POD Gold (GOG, US$5.99) - gameplay
Screamer (GOG, US$5.99) - gameplay

Need for Speed III: Hot Pursuit (1998)
Need for Speed: High Stakes/Road Challenge (1999)

all other are crap ;)

on pc i find Grid and gas guzzlers extreme really good, if you have a console you need to try Ridge Racer, its amazing, just avoid the demo for the PSVita and Unbounded.

Need For Speed Most Wanted 2005, Lego Racers 2, GRID 2, Disney Pixar Cars (Yes i like it)

The Crew (Uplay), GRID Autosport (Steam), Burnout Paradise: The Ultimate Box (Steam/Origin)

Grid 2 is (so far) my favorite.
It's great, beautiful, and has a great selection of cars. You get to race all over the world, both on roads and tracks.

Going by the referenced titles, I'd say you're favoring arcade racers.
So I'll recommend Split/Second. It's a bit like the Burnout series, the difference being that the power bar you build isn't used for simple speed boosts. Instead, it's used to set off opponent-wrecking and possibly track-changing Michael Bay-style explosions.

The Crew

Old school eh?
Midnight Club 2 http://store.steampowered.com/app/12160/
Flatout http://store.steampowered.com/app/6220/
SRS http://store.steampowered.com/app/292410/

Mobil 1 British Rally Championship

I don't know if it's similar to Asphalt or Need for Speed as I've never played those titles, but I think it differs quite a lot. I remember playing it years ago and enjoying the heck out of it! You may think that the graphics are a bit dated but I still love them! It released in 1999; yep, you read that right, 14 years ago. But it aged really, really well. Certain old games feel really old and ugly, but this one definitely doesn't. In my opinion, it still blows and watching a video to remember what the graphics looked like makes me so want to play! :D
The only downside is that it's not easy to get it to work on newer systems, and sometimes it crashes (not randomly though, it'll crash at the same place).

Make sure you have a racing wheel to play, it's way better, feels more realistic and precise. I recommend the Thrustmasters, because I had one and it was great!

Blur

For something arcade-y like the NFS series, I really enjoyed Burnout Paradise. For something more technical, I enjoyed Dirt 3. And for something more about racing real cars on tracks, I'm really looking forward to Project Cars. Gotta have a racing wheel for that last one..

Assetto Corsa or Project Cars

My favorite racing game of all time is ATV Offroad Fury: Blazin' Trails.

The ones I have enjoyes so far: Burnout Paradise, Sonic All Stars Racing Transformed (and the previous one too), Fuel, Grid 2.

Dirt Showdown Very much arcade feel to it, feels much like the Need for speed underground games. Very fun, and can usually be picked up for very cheap.

any Forza

It's not realy like what you are looking for but I highly recommend you to check out TrackMania2. You can download the demo on steam and play pretty much the entire game for 72 hours. Also there is 3 different version of the game: Stadium, Canyon and Valley