Probably less than you think. Each competition we do I find that we bring less as we realize that we don't need 3 rolls of paper towels, 4 boxes of gloves, etc. We cook out of a van or truck, depending on the event, and space is at a premium. Make sure to have a list of what you need at a minimum and go from there. If you don't have a trailer, the worst part of a comp is unloading and packing back up, so try to make that easier on yourself.
Then, take notes at the comp on what you used, didn't use, and what you wished you had. Refine your packing list for the next comp.
I'm still guilty of packing too much. I bring two sets of tongs to every comp and I can't remember the last time I used even one pair. If you can, do a practice cook at home and take notes on what you really need and what you can live without.
That said, a few things I've found valuable are: extra clothes to change in/out of as the weather dictates, clock/watch to keep up with turn-ins, more fuel than you normally use just in case, more beer than you think, advil, antacids, first-aid kit, and a charger for your phone. Other than that, you should be good
Nothing is worse than running out of beverages, except perhaps not having the meds to help out Saturday morning.