Technically Drving games... just not with cars!
I have a number of Train Driving Simulation games! I have these on the PS2, PSP, PS3 and one on the Neo Geo Pocket Color! All except one were NTSC/J only releases.
There are two series of games. Densha Da Go ("Let's go by train") and Train Simulator Real. Densha games relied on computer generated images. Train Simulator uses actual video footage which could be speeded up or slowed down according to the speed you were driving at. So, the PS2 can make claim to having the first photo realistic games.
These games are difficult. The Train Simulator featuring a Japanese High Speed Trains is the easiest. You have to obey speed limits, get to the next station on schedule and stop in the right place!
The High Speed Train game also has a slower train. Difficult to get very far with as too many faults is game over. Demo mode with that train gives a nice restful 50 minute or so journey through the Japanese countryside to the coast. At least I get to see some of Japan!