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TGlad:
This isn't fractals, but here's a post of my thoughts on average polygons:
https://tglad.blogspot.com/2020/11/mean-polygons.html

I thought you might like it.

It might extend to fractals... perhaps it is possible to find an average limit set, or average fractal curve. Who knows !  :P

Linkback: https://fractalforums.org/index.php?topic=3884.0

TGlad:
I've also had a quick go at mean polyhedrons: https://tglad.blogspot.com/2020/11/mean-polyhedrons.html

Well actually I didn't get beyond the tetrahedron and cuboid. It gets tricky after that.  :)

TGlad:
More generally, I've had a look at what are the average repeating signals:
https://tglad.blogspot.com/2020/12/mean-signals.html

still a few variants, but below is probably the best. They are mean repeating signals (signed numbers) of length 1 down to 8:

TGlad:
The equivalent in 2D is average textures: https://tglad.blogspot.com/2020/12/mean-textures.html

Very slow to converge, so I'm only able to go up to 5x5 (and that is a bit dubious)

The greyscale is 2x2 tiled texture up to 5x5 (right). Signed values, offset so grey is 0.
The coloured is complex valued, with real value in red, imaginary in green, and offset so grey is 0.

TGlad:
Moving to continuous symmetries, what about average sets of points: https://tglad.blogspot.com/2020/12/mean-points.html

In a line they follow a quite simple pattern:

In a sphere it seems less simple. Here it is for 5 points:

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