Every single non-cosmetic part you took off your car. Seriously. We ran into overheating problems with our revised setup and had to go back to stock - which was back at the garage. Fortunately it was only 45 minutes away and not all our team had even left there yet.
Every tool you own including ones that don't have anything to do with your car. You can barter them to other teams that need them. Don't forget things like a sawzall and come-along for emergency bodywork.
If you have any power tools, bring a generator just in case you can't get track power.
Every pair of clean underwear you own - because that goes in the race suit with you and gets turned into a sweat-soaked rag that you totally do not want to wear when you change back into real clothes.
All the air handling gear you can swing - at least one fan you can point at the radiator when it's in the paddock, and some stuff to point.
A shelter of some kind. We have a BIG tent that the whole car (and the whole crew) will fit under for shade. Something that just the front half of the car will fit under is plenty, though. This is especially important if you aren't bringing an RV.
If you intend to use your enclosed trailer as anything other than parking and storage space, INSULATE IT AND AIR CONDITION IT. PERIOD. We had an air conditioner running into our (black, aluminum-roofed) trailer and no insulation - if we hadn't had as much water as we did, there would have been hospital trips. You need both.
Speaking of water and hospital trips... Bring as much water as you can carry. We had 8 cases of non-booze liquid for 10 people. It was barely enough, and wouldn't have been if people hadn't gone home overnight. Water is both cheap and life-sustaining. Don't cut corners.
That said, warm water sucks. Fill every cooler you can find with ice, and figure out where you can buy more nearby.
A tow strap of your own, and a vehicle to tie it to. You DON'T want to have to push your car all the way across the paddock by hand.
PHONE CHARGERS. A lot of these racetracks are out in the middle of nowhere and even if your phone has 5 days of battery life, it'll probably drop to less than a day as it amps up its transmission power in a struggle to stay in contact with the network.
Driver, Pit Monkey, Rod Buster and Engine Fire Starter
Team FinalGear