Which production tools do you consider to be essential to your work? What do you firmly put in the nice-to-have category? I'd throw a sturdy tripod in the first category and a high-end mic in the second.
I agree with you wholeheartedly about bringing along extra batteries. I also make sure to bring extra extensions and charging cords because you never know what will happen and those are fairly inexpensive things to pick up that will save your project when you need them.
A must-have when it comes to videoing or photographing weddings is a second shooter. I agree with the rest of you guys that extra batteries are important and I think extra photo/video storage is important too. I'll use a lighting kit for engagement photos sometimes, but it's not like I'm dragging the kit to the wedding venue so I consider that to be a nice-to-have. Props fall into the nice-to-have category for me.
A tripod, extra batteries, photo storage, a mic, and headphones are some of my must-haves. You can do a lot with just that equipment and a camera. Some nice-to-haves is access to expensive software like Adobe Lightroom.
I understood the importance of custom lighting after being given a difficult assignment. I hired it then but bought it a few months later. A robust set of lighting equipment is essential, especially to those who specialize in wedding or portrait photography.