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/

These days it seems most pages are more JS and CSS than HTML...

Well, you know how long you've been hearing about Servo, and how it still, to a first approximation, can't render the web? Yeah, they're also bringing up Rust next to it, but... yeah, that's how long it takes.

It wasn't easy ten years ago, and a modern browser is at least an order of magnitude more complicated (WebGL, more complicated CSS, JS support, heck I'm not even really counting JS JIT here...).

Not really maintained recently, but there is:

https://common-lisp.net/project/closure/

There is Cocktail written Haxe (can compile to C++):

https://github.com/silexlabs/Cocktail

Microsoft uses its own EdgeHTML in the new "Edge" IE

I wish they'd open source EdgeHTML. I suspect it's far lighter than Gecko or WebKit, given that its development basically started with “let's rip out half of Trident”.

It probably only compiles on Windows, though.

There's h-smile (commercial) from Terra Informatica, that's used by a lot of antivirus vendors to implement desktop GUI software in HTML. The latest commercial branding for it is called Sciter.