Depends on what you mean by independent. If you just mean not any of the big three and their derivatives, you've got Servo[0], although that's still a herculean effort with direct support from Mozilla in terms of paying people to work on it. On the other end of the spectrum, you've got things like Dillo[1], which don't support much modern HTML/CSS/scripting, but do seem to get on by with far less effort.

[0] https://servo.org/ [1] http://www.dillo.org/