There have been other similar questions before (e.g. What is your picture of the flat topology?), but none of them seem to have been answered fully.

As someone who originally started in topology/complex geometry, the étale topology makes some sense to me. It's sort of like the complex topology, in that there are enough "open sets" that things like the inverse function theorem work.

But I don't really understand what these other more "exotic" topologies represent. From the (naive) background of topology, it sounds like we're just "adding more open sets" to our topology (which I know isn't right, since this isn't a topology in the standard sense).

So what do they represent? What do they do for us?