The obvious things you need for training jiu-jitsu are a gi, a belt and some underwear, plus it's traditional to turn up and go home with clothes on so some of them too. But beyond that there isn't really much else which are totally necessary. However, there are quite a few things I recommend you invest some money in...
Ice packs - get at least two and keep them in your home freezer. Using ice packs on minor injuries (I do it 20mins on, 10mins off as long as possible) will help you to heal much quicker and it's such an easy thing to do.
Spray-on plaster - normal plasters won't stay on due to sweat, even masses of tape can come off during hard rolling and regular training quickly uses up a lot of tape. The spray-on plaster you can buy will prevent bleeding and I've never found it to fail, although sometimes I have applied 2-3 layers (knees/toes).
A proper gumshield - I definitely don't mean a cheap boil & bite. I never understand why people will spend money on "limited edition" gis and top end custom rashguard/shorts but not on a good gumshield. I have an OPro and highly recommend them. These sort of gumshields cost over 10 times more than the cheap ones, but they fit perfectly and can save you huge dental bills.
Alcohol based hand spray/gel - an easy way to at least clean your hands before you head home (obviously not necessary if your academy has showers) and also helps to clean any wounds you open up during training. You could also go for packs of disposable wipes, but I don't find these as easy to use.
Electrical tape - great for taping up toes or fingers, or covering cuts if you don't have spray-on plaster. Also works well to sort out numerous other problems.
Kneepads - just the thin variety, nothing major. Your knees take constant punishment during jiu-jitsu so reducing the problem on impacts is worthwhile.
Cold spray - good for any minor injuries you pick up, before you get home and can ice it.
Heat rub/oil - helps to loosen up any muscles which are tight/sore before training. Don't use too much though, and only apply on areas which will be well covered.
Kitchen roll - good for wiping up any blood from you or your partner and it doesn't fall apart from sweat like toilet paper/tissue.
Small towel - goes without saying that you will be sweating a lot during training. Wash it every time!