Running inverters of any size off a cigarette lighter plug is a dangerous game (at least when considering the demands that chasers typically need). Many inverters come with a set of cables that allow you to plug it in cigarette lighter, but that doesn't mean it's a good idea. They sometimes go bad, as many have found out, and start blowing fuses. The real problem is when the fuse DOESN'T blow. I've had a situation in my personal car where the socket actually shorted before the fuse and welded itself together, ruining the entire socket. I ended up having to replace the whole thing which is a pain in the ass. Bottom line, if you need an inverter find a way to tie it to the car battery.
What I do is run a line of large cables (8-10 AWG) to the battery and put in my own fuse box inside the cabin somewhere. I also put a switch in line so I can turn the power on and off. If you run large enough cables, you can run a much larger inverter inside, say an 800 W inverter. This would give you enough power to run almost anything you'd need (say and xbox and a 46" LCD monitor to play it on), without having to worry about blowing any of the components hardwired into the car. If you do something like this, be sure to put a fuse as close to the battery as possible to prevent the wire being nicked somewhere along the way and directly shorting out your battery.
