The things I used most, or wished I had the most:
-Drawers, crates, or boxes. Like a PP stated, you're sharing closet and bathroom space, and especially in the converted 2 bedrooms (3/room/bathroom), storage space can get a little tight. I brought different-sized plastic boxes and crates that I used for different things, in addition to my closet and drawer space.
-Seconding the recommendation for a good backpack. Even if you don't use one to play in the parks (I did), you'll want it for work. Bonus points if it's large enough to fit a change of clothes so you don't have to stop at home before going out to play.
-Lunch box. Depending on where you're working, food options aren't always conveniently located to get to during your breaks, especially during busier seasons. I packed my own lunch almost every day working at AK. I'd go for a bigger lunch box (to go in your bigger backpack!) that you can keep an extra drink or snack in for longer days--on my 10-11 hour days, I usually needed something later on to keep going, too.
-A lamp. I lived in Chatham, and we had overhead lighting, which was great, but my roommate and I were on totally opposite schedules. Mine was a cheap desk lamp from Target that gave me just enough light without bothering my roommate.
-Some cooking tools depending on what you like to cook. I brought a set with a cookie sheet and casserole dish, and since I cooked more at home, it was nice to have. I really wish I had brought a Crock-Pot with me; I probably would have cooked even more at home if I had had one, especially on longer days.
-Tupperware or similar food storage. I ended up buying a lot of cheap dollar store Tupperware, which worked perfectly fine. Used all the time for leftovers, lunches, etc.
-Cold weather gear, no matter where you're from or how used to cold you are. I'm an Iowa girl, and despite having been to Florida in the winter before, I didn't think I'd need much in the way of sweaters or jackets. That was entirely untrue; we had some very cold days, especially in my first couple weeks. Furthermore, I'm of the opinion that you acclimate to the weather quickly, so warmer temperatures might seem colder to you after a while even if you're from a colder climate. Sweater, fleece jacket, jeans were all worn in the time I was there. I wish I had had gloves or a warm hat with me, and ended up buying a cheap pair of gloves while I was there.
That's everything I can think of right now, but I'll add if I remember anything else...