The Tree Style Tab extension[1] is wonderful for "keeping tabs" on our modern "to forget" lists (unclosed tabs). Putting tabs on the left is also a good screen real estate trade for wide aspect ratio screens.
I use Sidebery, which is similar, but adds features like 'panels' (essentially switch between sets of tree-style tabs in the sidebar) that integrates with Firefox 'containers', so you can have a 'work' vs. 'personal' panel&container for example.
The tabs being listed vertically is by far the most important aspect to me though, I previously used something (that stopped being maintained) that did that without being 'tree-style'. Nesting is probably not what I wanted as often as it is.
Tree Style Tabs is absolutely vital to me, I don't understand how people can live without that extension.
The other thing I use a lot is firefox's readability builtin (it's not an extension), which removes all the cruft and *walls from websites and makes them... readable in one click.
These two together are killer apps for me. I might consider switching to another browser (firefox isn't exactly the paragon of virtue it used to be either anymore), but the new browser would need to have these two features.
It's the hierarchical tabs that I need. If you are traversing large hypertexts you need to keep track of where you are and where you were.
Middle-clicking to open-in-new-tab together with Tree-Style-Tabs is incredibly useful in this scenario.
You can rapidly open a large number of potentially-interesting links in tabs, and then weed out the ones you don't need just as rapidly. Then wash, rinse, repeat. It's an assembly line approach.
As someone who uses wide screens to hold multiple page-shaped windows, sidebars are a no-go. I really like Chrome's recent addition of what they call tab groups; anyone know of equivalent functionality for Firefox?
Also there's a bunch of userChrome css people have shared for Tree Style Tabs that makes the sidebar collapse and auto-expand on mouseover so it doesn't take much space. It's not part of the extension itself because they lost access back with the Firefox Quantum release and I guess no new API was ever made.
[1]: https://addons.mozilla.org/en-US/firefox/addon/tree-style-ta...