Depends on what camera you're buying--point and shoot? bridge? crop sensor? full frame? I'd say the "essential" accessories differ for each. Your own shooting circumstances are important as well--are you going to be doing portrait photography? nature and wildlife? sports? and on and on...
Since I don't know enough about YOUR situation to make any inferences, I can only tell you what *I* did. When I bought my Nikon D5100 last year, I got: extra battery, cleaning cloth, a circular polarizing filter, a case (actually ended up buying two--a small one to carry around on a daily basis and a larger one to take when I go out specifically to do photography), lens hood, wireless remote shutter trigger (which I HIGHLY recommend). Can't think of anything else I got with the camera (already had quite a few memory cards).
Since then, the most "essential" items I've added are ANOTHER extra battery, two additional (better) lenses and an off-camera flash.
EDIT: Oh, and I also already had a cheapie tripod that I used for my Powershot. But if you don't have one already, and you're buying a DSLR, don't bother with the cheapies. This is STILL on my "must get this soon" list.