Changing potentiometer from a gaming steering wheel
It's easy enough to get multi-turn pots, and they aren't all that expensive.

As has been pointed out, if you go this route, you'll lose resolution - that is, you'll need to turn the wheel more in order to get the same effect on the game vehicle.

You may well have mechanical problems. The existing pot probably has an intrinsic range of ~330 degrees, but the wheel mount has built-in stops to keep the 270 degree range, and do it so that the pot doesn't provide the limiting (that would be a great way to destroy the pot). You would need to modify the wheel assembly to get more rotation. Once you do this, there's no easy way to build in stops for multi-turn ranges, so limiting would be done by the pot. That is, if you were trying to make a very tight turn, you'd turn the wheel until the pot ran out of range - and I can pretty well guarantee that you'd kill the pot. These things just aren't that strong.