Hmm... probably a couple of days or so. I've done this quite regularly so it's hard to pinpoint what the longest time was, plus I can go for a scarily long time just drinking coffee.
At my last job I was basically living in less than a meal a day because I didn't have time to eat. By the time I finished work I was completely shattered and had to be back at the wretched place in 5-6 hours so needed to snatch all the sleep that I could, so understandably couldn't be bothered cooking (especially since that's what I had been doing all day).
Also, getting messed up on drugs does nothing for your appetite, unless you're smoking weed.
Now I am, by comparison, a much healthier human being.