CrossFit is great because for the most part it is a minimalist program. Yes you can do CrossFit with expensive gear like GHD machines, rowers, olympic barbells and bumper plates, etc., but you can also do it with your body weight, a place to run, and some playground equipment.
That being said there are some pieces of equipment that you should seriously consider investing in to make your CrossFit training better. They are by no means necessary, but having access to them will improve your training and your fitness.
1. The first on my list of recommended equipment everyone should have is your own jump rope. How frustrating is it to come to the gym for a workout involving double unders and not be able to find a rope your size, or none at all? A jump rope is a personal piece of equipment that is inexpensive, and makes your CrossFit experience better because it's made for you. My personal favorite rope is Rx Ropes. They will size the rope to your exact specifications, and they have five different cable weights based on your current ability, which allows for progression as you become more proficient. Rogue Fitness and Again Faster both makes styles of ropes that many people like as well. I'm not endorsing one product over another, but get yourself your own rope, sized for you, and watch your double unders improve and along with it, your fitness.
2. Wrist wraps are another piece of equipment that should be in everyone's gym bag. Most of us don't think about adding wrist support until we've hurt ourselves somehow. I personally use wrist support on most pressing movements when I get above the 75%-ish range just to keep my wrists healthy. We tend to support heavy loads with our wrists in extreme extension (such as the clean, the front squat, the thruster, etc) and some added support will only help ensure health and longevity in your wrists. $12-$18 is cheaper than buying lots of tape, and it won't give your wrists and forearms a wax job every time you remove them. They're cheap, effective, and and will protect your joints. Just get some!
3. The next item I would recommend is a weight lifting belt. These come in many different options: thickness, width, velco or buckles, leather or nylon, etc. Find one that you like and buy it. Lifting with a belt should not be a crutch for a weak core, but it can protect your back when attempting 1rm deadlifts, squats, cleans, etc. Your belt should allow you mobility to get into a deep squat position so, because our body types all vary so much, having the perfect belt for you at the gym is unlikely. Spend a few bucks now and save your back while you're getting after it. You can thank me later when you're getting after it and setting PR's!
4. The fourth and final piece of equipment I will recommend is a good weightlifting shoe. When you bowl, you wear bowling shoes. When you run, you wear running shoes. When you want to be a hippy and look ridiculous, you wear Vibram Five Fingers. Whatever activity we do in every aspect of our lives, we wear the appropriate corresponding footwear. So why is weightlifting any different? Weightlifting shoes provide a firm, sturdy platform for your feet, giving you maximum power transfer to the load you are lifting. They help put your feet, ankles, and knees in alignment for proper movement and mechanics. The slightly raised heel helps you activate and load the muscles of the posterior chain for maximal strength and power. Weightlifting shoes give you the perfect combination of safety, efficiency, and strength potential that lifting in running shoes will not. Reebok, Adidas, and Nike have all developed weightlifting shoes that are also fairly flexible to allow you to run, box jump, jump rope, do burpees, etc. so that you can keep them on during met-cons if you like. Weightlifting shoes generally will run between $90 and $200, but they are pretty much a one-time investment. I have met Oly lifters who had the same pair of lifting shoes for over 10 years. I personally have have mine for 3 years and the only thing that has worn out has been the laces. Do your lifts a favor and get some real lifting shoes, and for the love of all that is holy please don't olympic lift in Vibrams. It's just wrong, and we'll talk about that later.
There are other pieces of equipment and gadgets out there that are nice to have, but are not essential. Many people like to have their own set of collars for their bars, a pair of gymnastics grips, compression clothing, neoprene sleeves, weighted vests, etc. They all CAN be effective tools, but you can also get into excellent physical condition without them. The same goes for the items I listed above, but I find them very useful, convenient, and beneficial to add to your gym bag. If you really want to take things to another level, think about investing in yourself and pick up one, two, or all of these items.