Of all the car racing games I have spent the most amount of time time, the free-to-play Need for Speed World could have been considered my second home. Even with the online servers shut down for over a year now, I still find myself going back and taking in some old sights and singleplayer action.
Customization of vehicles was only limited to a player's skill and time dedication to achieve the look they wanted (the creative liberties were only limited by the imagination). Upgrades could be earned and applied, giving immediate results for the positive or negative. Although the game never achieved a truly completed status, there was more than enough things to do, places to see and tasks to perform that would have allowed the game to be running online to this very day.
Including some police action.
At one point in time, a player could just cruise around and tap a roaming police unit and the chase would be on. I can't remember which update or when it was released that removed this option from the game, but I do miss it. Sadly, if a player wanted a light show from the police in their rearview mirror, a mission has to be selected for the fun to begin.
Perhaps, calling it "participating in insane action" might have been a better phrase for it....
At nearly the same time in 2010 and on the paying part of the spectrum, Criterion Games released a re-imagined and rebooted version of the classic title Hot Pursuit. Much like their latter Most Wanted, customization was limited to vehicle paint choices and upgrades just didn't exist (although vehicles would be equipped with devices like spike strips and EMPs, those really don't count nor carry over past the missions they are used in). All the cars handled very differently from World, tires lacked serious grip, causing the vehicles to be prone to easy sliding.
The upside. I found my old copy of the game collecting dust in a box and installed it. After locating and restoring my old saved game file, I rediscovered the Lamborghini Murciélago was a select-able vehicle for player use (seems the police would try and counter me with their own version).