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It is easy to notice the fractal STRUCTURE of things in the universe at different scales (like a solar system resembling an atom), but I was just researching the actual SIZE of things at different scales.
In particular, starting with a small building block (the atom) and seeing how many it takes to get to another "building block", then how many of those it takes to get to another, and so on.
I gathered some numbers just from google searches and wrote them down and see if there was a pattern. I put down estimates from as many sources as I could find to try and get a decent average -

Number of atoms in a cell -
10^14 - 10^16

Number of cells in a human -
10^12 - 10^16
3.72 * 10^13
4.6 - 6.8 * 10^16

number of atoms in a human -
7 * 10^27
2.3 * 10^28

number of people shoulder to shoulder that can fit on earth -
5 * 10^15
(didn't see any other info on a sort of "human to earth" scale)

number of earths you can fit in the solar system -
3.3 * 10^27

number of solar systems you can fit in the galaxy -
I couldn't find any estimates on this so I did my own.
I came up with 8 * 10^15
With the milky way being 100kly in diameter and 1kly thick, you get about 7.8 * 10^8 cubic light years.
I used the orbit of neptune for the diameter of the solar system (0.000963 light years)

The numbers here are probably under big assumptions and hard to calculate for sure (the edge of the solar system could be calculated as all sorts of different things)
But I did notice the exponent used for every step up in these estimates is roughly a multiple of 14.
And something odd is that if you assume these numbers to be correct, you see a missing spot in between the "earth scale" and the "solar system scale"
I wanted to continue on to something like "the average number of galaxies you can fit inside an average sized galaxy cluster" but it starts to get more and more general and I don't think you could come up with anything accurate.