<p>What is the Emacs mode or package that highlights Lisp forms changing the color of the backgrounds so that the form you are in has one color, the outer form another, the outer outer form another and so on?</p>

You may want to try
mwe-color-box
(screenshot below) or read
Five approaches to s-expression highlighting
by
Lemondor
.
(source:
foldr.org
)