I've always assumed that the occasional disfiguring burn was part of growing up.
I know my own curiosity got me injured several times as a kid.
One time, while either my sister or my mom was getting their hair done, I grabbed the curling iron with my bare hand. Needless to say, this resulted in a rather severe burn. I didn't get taken to the doctor or anything, I got given a bag of frozen peas to hold.
Another time my mom was shopping and I was being taken along. On the checkout counter in one store there was a candle in a jar. I was curious, so I picked up the candle, and tipped it over so I could see inside, spilling molten wax on myself. Again I think I was given a bag of peas.
I can't remember any of the trauma that comes with these burns. There was no lawsuit, no doctor visits, nothing. I can't even tell you where exactly I got burned. And I wasn't too much older when I did these things to myself.
2 years ago I laid my wrist on a fresh weld, which had just cooled enough that it wasn't glowing anymore. I had an impressive second degree burn across my forearm, and had a bandage on it for a few weeks, and got myself a bag of frozen peas to hold against it. Now I can't even remember which arm it happened too, as I can't find a scar.
For injuries such as this, where it is more than likely the child's own fault, I don't see any need for compensation beyond medical bills (all I got was a few bags of frozen peas) and I highly doubt that there will be any permanent damage resulting from this.
Now what I did to myself with a disc sander left a much more interesting scar than my multiple instances of getting myself burned.