How can I append to an empty JSON object?

It sounds like you want to end up with a JavaScript object containing several instances of objects that have two keys value and children. It would seem an array is the best choice (which Khnle's and Chris's answers give you):

[{"value":2,"children":3}, {"value":12,"children":9}, {"value":20,"children":13}]

In your comment to one of the answers, however, you said that you did not want an array. One way to do this is to wrap it, as in Jergason's answer:

{
    "children": [
        {"value":2,"children":3}, 
        {"value":12,"children":9}, 
        {"value":20,"children":13}
    ]
}

Your question seemed to say that you like arrays because you get the push operation, but you would like to avoid them completely. The only way to avoid arrays completely is to tag each object with its own unique key. If indeed this is what you want, it would look like this:

{
    "child0":{"value":2,"children":3}, 
    "child1":{"value":12,"children":9}, 
    "child2":{"value":20,"children":13}
}

This is not hard to do; just replace kids.push(child) with kids["child" + i] = child.

Make sure this is really what you want, though, because this collection of children really seems to scream "array"! :-)

You could just have your object contain an empty array.

var obj = { "children": [] };
// Your looping code here
obj.children.push(child);

With some slight changes (add var for local variable):

var kids = []; //Note this line
var i = 0; //index counter most likely needs to be local
while (i++ !== count) {
    var child = {
       value: Math.floor(Math.random() * 100),
       children: addChildren(Math.floor(Math.random() * 10))
    };
    kids.push(child);
    console.log(kids);
 }

I think that's what you would want.

Update: Your requirement is rather odd. You can do this:

 var kids = [{}]; //Note this line
 delete kids[0];
 var i = 0; //index counter most likely needs to be local
 while (i++ !== count) {
    var child = {
       value: Math.floor(Math.random() * 100),
       children: addChildren(Math.floor(Math.random() * 10))
    };
    kids.push(child);
    console.log(kids);
 }

or

 var kids = [{
       value: Math.floor(Math.random() * 100),
       children: addChildren(Math.floor(Math.random() * 10))
 }];
 var i = 1; //1 less iteration than previous
 while (i++ !== count) {
    var child = {
       value: Math.floor(Math.random() * 100),
       children: addChildren(Math.floor(Math.random() * 10))
    };
    kids.push(child);
    console.log(kids);
 }

That would satisfy your requirement, but what I show originally is still, I think, what you want.