Not sure what you consider "production" but suffix arrays et al. are used fairly widely as far as these data structures go, but they aren't particularly exotic. Curious to hear about some more exotic ones though.

I use graphs because we have a case where we need to schedule actions/programs but they themselves have "ingredients" or dependencies in a wild arrangement. So a topological sort of a graph is incredible to solve this and run those actions in order.

I don't think just plain graphs with topological sorting are considered advanced data structures? The're one of the first things you learn in a CS curriculum... whereas lots of the stuff in Demaine's course is stuff that pretty much doesn't even exist in most schools.

I mean I suspect we all have through compilers, standard libraries, and database query engines.

Succinct data structures are used in bioinformatics (e.g., in combination with FM-Index).

Yes. I work on compilers and data structures can make or break the case for implementing several optimizations.

Which one would you point out as the most useful from your point of view? It's much easier to learn that topic looking at the practical side as well as the theory.

Representing integer ranges and working on dynamic graphs are everywhere in compilers. Also hashing and string algorithms, for obvious reasons.

Which ones from that course did you use?

what... hash tables? :) yes