I have some excessively exposed images that show tremendous improvement in Lightroom Develop when I drag the Basic "Highlights" slider all the way to the left (to -100). What I don't understand is why it stops there. How can I get even more of the effect it is producing?
The sliders only work within a specific range where they can presume usable information within the image. Beyond that, there's a good chance of damaging the image by pushing the values too far.
That said, with images like yours, sometimes you need extreme measures. There are a couple of ways you can push the limits here. The easiest is to switch to the Tone Curve panel, where you can also adjust the whites and highlights, and that effectively allows you to double the change that you can make in the basic panel.
The other way to do this is to use either the graduated filter, the radial filter or the adjustment brush, cover the area (or the entire photo) with a layer, and then make adjustments there as well.
Note that doing this you're likely going to find it hard to get a GOOD photo out of it, but you may well get a usable one.
Lightroom simulates a photography darkroom processing, so the values of highlights, shadows measure in %, so you are applying a value up or down on the the highlight pixels in the image. If you want to add an extra effect you can combine with the exposure or the curves. Or for particular areas you can use the brush tool and apply more.
A great way to get over that limitation is at the CC Photoshop to apply that values and duplicate the layer and go to basic highlights and adjust again...in Lightroom you also can do it, replicate the image and process it again. But for extreme processing i recommend CC Photoshop or an equivalent.