Here is the gear I wear:
Custom mouthpiece: Everyone needs one (at least a boil and bite if nothing else)
Knee pads: I have had meniscus surgery on both knees, so the knee pads reduce some of the repeated impact with the mat when I'm on my knees (guard, turtle, etc.) They won't hurt to wear, but I would consider these optional for most people.
Gear I don't wear but may be recommended for others:
Cup: I don't wear a cup because it interferes with techniques. They are illegal in IBJJF competitions. They also have the potential to get pushed out of place and smash your nuts badly. Without the cup, I rarely get hit in the groin, and the hits you take in grappling are usually pretty weak. It is not like Muay Thai where if you get kicked in the groin full power you could lose your testes. Most BJJ groin hits are in the walk it off territory. That being said, I'd never stop anyone from wearing a cup in training if they wanted to.
Headgear: I don't seem to be particularly susceptible to cauliflower so I don't wear it. However, for the people who do have problems, this is a very important piece of equipment.