I want to export an illustrator file...
72 ppi does not save at high enough standard. 300 ppi makes the dimensions too large.
The end user should never use a file smaller than 55x20mm, so I want to give them the highest quality file possible at that size.
Thanks in advance!
To give the best possible quality file, and the exact size you have to send a vector based file. Svg or a pdf file (converted to curves or nodes) are two good options.
Asking a specific size in mm means that you need a printed file, so in that case the option to a rasterized file can be from 200 or 300 ppi for a rgb, cmyk or grayscale images to 1200 or 2400 ppi monocromathic files. In this cases the size you see on your screen does not matter.
If on the contrary, you need to see it on your screen you can not control the size on cm at all. Are you using a mobil phone or proyecting it in a movie theater? ... You simply can not.