How thick is a snowflake? The x, y dimensions of snowflakes get all the attention for obvious reasons, but the z dimension must be interesting too, in its own humble way.  In fact, I have lived more than half a century and never seen anyone refer to snowflakes' third dimension at all!

So I'd like to know what a snowflake's cross-section looks like, and what forces/phenomena make it that way.

There are numerous possibilities:


that snowflakes are a single molecule thick
single crystal thick then grown over with hoarfrost
thousands or millions of nearly identical snowflakes glued together like slices in a loaf of bread
something else


No doubt, the ratio between x and y dimensions is extremely similar across all snowflakes, but an interesting part of the phenomena involved would be that which defines the ratio between x and z dimensions.
There are many different types of crystals and their fromation is far too difficult for me to understand. This answer simply deals with the thickness part and that too merely gives a rough approximation due to the lack of data.
 From CalTech

   

We have two formulae for calculating the width:  


For Dendrite
$$\mathrm{h =a_2 D^f} $$
D: crystal diameter (cm) (major dimension)
h: crystal thickness (cm) (minor dimension)  
For Needles
$$\mathrm{d = a_2 L^f} $$
D: crystal length (cm) (major dimension)
d: crystal thickness (cm) (minor dimension)    


This is a table for $\mathrm{a_2, f}$ 

  

For d and l   (right most column )
   

I calculated the thickness of a $\pu{3 mm}$ Dendrite which is coming around $\pu{24  \mu m}$. 



Source  

https://web.archive.org/web/20180416083128/https://www.nsstc.uah.edu/data/uahsevere/owles/parsivel/references/Snowflake_Size_Distributions_Lecture.pdf