If you're okay with older games and want some arcade racing, then the old Microsoft Madness series are pretty good, especially Motocross Madness. None are on GOG though, and probably never will be.

As for driving without a wheel...I just cannot do it for anything with even a remotely realistic driving model. I just suck so bad at driving with a pad. I don't care about force feedback, so even the cheapest non force Thrustmaster wheel is 100 times better for me than using an analogue pad.
I still have my original Microsoft Sidewinder wheel from around 97' or so...and even though it's almost 20 years old it still works just fine for PC games, as the optical sensors with chopper wheels do not wear out like the old pots did with older joysticks/wheels. Only problem is it uses a midi gameport, and won't work with Gameport to USB adapters for some reason. So a wheel can be a decent investment as long as you're serious enough about driving games.

The price for the one I use today on PC was about €250 (+ €250 for the pedals), at the time of purchase, the Fanatec Porsche 911 GT2. I've had it for a long time without issues, and it works on most consoles as well. So a total cost of about €500 for a product I have already used for a long time and will probably be able to use for many more years, is a good deal for me.

At the time I also wanted a wheel that would work on PC as well as on both consoles, which is very rare, as most racing wheels only work on one or two platforms. So this particular wheel is quite unique even today. For example the later Logitech G27 is €400 more expensive than the GT2 but will only work on the PC and the PS2/3, not on the PS4 and any of Xbox systems.

I also use a gamepad that has a mini-wheel attached to it, I bought this one for the PC (big fan of DTM). It works with every sim game I play: RaceRoom, Race 07, Stock Car Extreme, Automobilista, rFactor, Project Cars and a couple of others. It even works well with the MotoGP and SBK superbike games. The benefit is that you get a much smoother steering than with a regular controller.

Grand Prix Legends -- not for the faint of heart, a hardcore sim.
NASCAR Racing Season 2003 -- also not for the faint of heart.
Power & Glory (mod for GTR2) -- great classic cars racing series, easier to get into but great attention to detail especially on physics.
Richard Burns Rally -- The king of rally games for the PC.
iRacing -- for your online sim racing needs (subscription model)

edit: you won't find them here. :P

POD and Wacky Wheels!

The Wipeout series was pretty cool. But it's unlikely we'll ever see the DOS/Windows versions on GOG...

MX vs ATV Unleashed is available here. It's a lot of fun!

Personally, I have no problems driving with a pad. I recently bought a Elite-controller, and need something to use its full potential on. Can you recommend some good driving games on Xbone? I'm considering grabbing a month of gold to get the first Forza Horizon.

Pretty much anything with Forza in the name. They are hugely popular for good reason, they actually are good. They are made from the ground up, and are fully optimized for the hardware and it shows in practice. Forza 6 is the best all round arcade racer ever made IMO, looks great is clear sharp and runs perfectly smooth and the cars handle pretty decent. But it's expensive and refuses to come down in price because of it's popularity. Even at it's current 40% discount it's still expensive...but it does have shitloads of content.
A good option if you can find a second hand disc at EB (or whatever you have in Norway) is Forza 5. It's a cut down version sure, but if you get it cheap enough it still has plenty of content and still looks and runs almost as well as 6. Horizons 2 is also a good second hand option for open worldness driving.

And if price doesn't matter then get Horizons 3 later this month, you can almost drive past my house if you get that one.

All the newer games also support the impulse triggers as well...which really works for driving. When your brakes lock or wheels spin under power the triggers vibrate to give feedback so you can ease off, very useful if using a pad.

Insane the old one, not the shitty remake :P and Need for Speed: Porsche Unleashed <3

If you're into arcade style racing games, You'll probably enjoy Trackmania. I recommend starting with Trackmania Nations Forever. It's not available here, and you'll need to create an account. On the other hand it's free ;-)

You can get it here: http://trackmaniaforever.com/

The download link is at the bottom of the page.

EB in Norway? No, but we have Gamestop (though their prices are shit), and there's a site for buying and selling used items, where I buy most of my console-games. I've seen the GOTY editions of Forza 5 being sold dirt cheap there, though it didn't get very good reviews when it launched. Much of the critique was because of micro-transactions. I assume all the DLC is in the GOTY edition though? I'll keep an eye out for Forza 6 and Horizon 2 as well. I played their demos a while ago, and it was quite nice! And they really never drop in price in retail stores.

Rollcage is the best racing game, too bad it's not on gog

Thanks for the suggestions!

Online game with custom levels?
This is nice, I'll give it a try.

Humblebundle currently have Project Cars at 66% off in their late summer sale.

Also Ubisoft Club are giving away The Crew on September 14th (admitedly a driving rather than racing game).

I'd suggest Grid Autosport for a nice mix between arcade and sim racers. That and Shift and Shift 2.

I also have F1 2013 that I'm really enjoying ATM.

MegaRace 1+2

It was a funny racing game on its time :)