* Palemoon (Goanna engine, derived from Gecko but forked a while ago)
* Netsurf (Independent engine, mostly good for basic HTML pages)
* Pre-2013 Opera (Proprietary Presto engine, no longer maintained, never open sourced)
* IE11 and before (Proprietary Trident engine, going EOL this year, never open sourced)
* Text/terminal browsers (lynx, w3m, etc, independent but only support text in the terminal)
Of those, Palemoon is probably the only contender as an independent browser with open source code that can support at least parts of the modern web (although the experience is rough, Palemoon was forked when Firefox still ran as a single process, meaning a bad tab can crash the whole browser, and web components don't work).
A modular browser would be nice - where custom components can be swapped in for certain functionality (layout, networking, JS engine, CSS engine, renderers for various MIME types).