How to fit an artboard to an object with a drop shadow?

In the latest version of Illustrator there are still some issues if appearance attributes from the Effect menu are applied at higher object levels (Group level (including clipping groups), Layer level and Sublayer level).

In those cases the additional spaces are still ignored by the Fit to Artboard Bounds or Fit to Selected Art commands.

It seems that all the answers here are governed by different (lacks of) version assumptions.

As Ton already said, in the latest versions of Illustrator it should work with or without the Preview Bounds option.

It used to be an issue in some older Illustrator versions, as far as I can remember.

I thought about the Preview bounds too, but even without them I get the desired behaviour, in 2022 and 2021.

Have you tried to Object>Expand Appearance? It will make the drop shadow basically a separate rasterized object grouped with the original shape. It can be selected and therefore you should be able to select it as what you want to fit the the artboard.

I would make a copy/paste in place of the object and rasterize it. With the rasterized copy still selected, do your Fit To Selected Art, then delete the copy. 

BUT

I just tried it in the latest version and Fit To Selected Art doesn't chop the shadow. What version are you running?