Since capacitance is proportional to the area of the plates, why don't we just make a mesh out of metal wire (plate #1) and coat it with a thin layer of dielectric material, then submerge this mesh into an electrolyte (plate #2) which is connected to one not-submerged, not coated end of the wire mesh? All that crazy big surface area would serve as the plates area!
Yes, you can of course use a wire mesh as a capacitor plate to increase the available surface area (compared to a regular metal plate).
The problem is that this is still extremely inefficient - the surface area of a wire mesh is quite low compared to the structures that are already in use in modern capacitors.
If you push things further, you'll eventually arrive at the conclusion that you want a very fine metal sponge. The surface area of such a sponge is immense (it has a huge number of connected pores in it), and if you can coat all of that surface area with a dielectric, you have an absolutely amazing capacitor.
This idea isn't new, and it's in fact how all modern tantalum electrolytic capacitors are made. Tantalum powder gets sintered into a metal sponge pellet with an immense surface area. This is then coated with a thin dielectric layer via an electrochemical process.
Aluminium electrolytic capacitors are similar: The electrode foil gets etched to create a sponge-like (porous) structure on its surface.