Without other context, 18.
…and it depends on the maturity of the kid / not-a-kid in question.
I’ve known 20-year-olds who were adults, and 30-year-olds who were still kids.
Although in my circle, after a certain point (and definitely once they are in their 30s) the term is less likely to be “kid” and more likely to be “big fat man baby.” 
“Other”
I got it from Heinlein. It’s the moment when you realize life isn’t necessarily fair.
You stop being a kid when you are responsible for your own food, your own transportation needs, your own rent or mortgage . . .
It’s a state of mind, more than an age.
When people start saying: “Here’s looking at you, you old bat.”
I haven’t.
In many ways I’m still a kid, but I stopped being a kid in a lot of ways when I had kids.
Growing old happens; growing up is optional. 
This may be a sixty year old body, but it’s still me in here!
Back when I was 24, I decided I was too “grown up” to go into a toy store. A few years passed, and I realized how stupid that was, and went right back to buying toys for myself.
Perhaps we’re really “grown up” when we get over being “grown up.”
If someone is using “Kid” as a dismissive term, I got pissed at 18.
Among peers, we were “kids” until graduating college.
But, for “Fully Grown Up”, QtM nailed it. On the great March to the Cliff, once your parents are gone, it means “You’re Next!”.
Or, as I was about to put it: At what age do you become a kid once again?
Kid ? Child = about 13 yo
A boy in my day was under 30 yo, all the old blokes would always call you boy, not in a nasty way, but it was all to do with respect.
When you start paying your own way. The only difference I’ve ever seen between kids and grownups is that grownups pay.
I recall being about 24 and getting somewhat annoyed at some guy on the Internet for saying everyone under 30 is a child. I was paying a mortgage, I owned a business with a dozen employees, and I was the only family member taking care of my terminally ill father. I definitely felt like an adult then.
I’m 32 now and not a lot has changed. My dad is dead and I have a wife and son. Marriage and parenthood are definitely new life experiences, but in terms of adult responsibility they’re actually a lot like taking care of my dad except extremely positive experiences compared to very negative.
One big difference is that now I couldn’t care less if someone thinks I’m done being a kid or not. Personally I still agree with 24 year old Fuzzy Dunlop, but he was mildly to moderately annoyed by some guy on the Internet and I wouldn’t even care if someone I knew well felt that way.
I voted “Other” because when my mother passed away, I said to my oldest brother “I guess we’re the grownups now”. I was 45 and had two kids of my own (he was 51 and had 4).
That said: I have to say if you’ve graduated college you’re no longer a kid in any meaningful sense. Other signs:
- You’re self-supporting
- You have kids of your own whom you are supporting.
I’ve got a mortgage, two car payments, a house windows payment, divorced and married again along with four children, two out of the house and I’m about to be a grandpa, that means I’m not a kid any more?
DAMMIT! I gotta quit playin hookie during the staff meetings and start reading those memos!
I voted other, as has been said, not a kid when you start being self supporting and supporting kids/others yourself. Will almost always be a “kid” compared to someone.
According to my mom, not til I die.
Story from my thirties: (on a speakerphone in a office of women)
“Well, do you have a coat? It’s freezing out there!”
" I listened to "Does your mommy think you’re cold? Poooohhh Bwasssyyy!’ for months…
From my forties:
“put this on, my god, do you even have a good coat?!”
I was wearing a wool sweater in Texas. And a pea coat.
Don’t even add the “do you need to eat that?/ Why aren’t you eating?” thing.
In my experience, it’s a relative thing. As you get older, the line gets love higher. At 31, I’m starting to think of any teen as a kid, when I used to separate the two categories.
When you start dealing with your problems as an adult. Talking instead of running away, for example.
For me, it was age 37. That year, it suddenly hit me that I was an adult. And I’d been out of my parents’ house since I was 19. I don’t recall a specific event as a trigger - it was just a slap of reality.