@Bergi mentioned new.target.prototype, but I was looking for a concrete example proving that you can access this (or better, the reference to the object the client code is creating with new, see below) without having to call super() at all.
Talk is cheap, show me the code... So here is an example:
class A { // Parent
    constructor() {
        this.a = 123;
    }
    parentMethod() {
        console.log("parentMethod()");
    }
}
class B extends A { // Child
    constructor() {
        var obj = Object.create(new.target.prototype)
        // You can interact with obj, which is effectively your `this` here, before returning
        // it to the caller.
        return obj;
    }
    childMethod(obj) {
        console.log('childMethod()');
        console.log('this === obj ?', this === obj)
        console.log('obj instanceof A ?', obj instanceof A);
        console.log('obj instanceof B ?',  obj instanceof B);
    }
}
b = new B()
b.parentMethod()
b.childMethod(b)
Which will output:
parentMethod()
childMethod()
this === obj ? true
obj instanceof A ? true
obj instanceof B ? true
So you can see that we are effectively creating an object of type B (the child class) which is also an object of type A (its parent class) and within the childMethod() of child B we have this pointing to the object obj which we created in B's constructor with Object.create(new.target.prototype).
And all this without caring about super at all.
This leverages the fact that in JS a constructor can return a completely different object when the client code constructs a new instance with new.
Hope this helps someone.