Hmm. Well.
There are biological things that have not been alive like honey and milk as above.
There are things that are synthesised but were originally organic, like vinegar.
Then there are things that are synthesised but were once extracted from organic things like MSG
As for things have never been biological in any way – quite a few in modern times of course, thanks to factory foods and so on. But I can only think of two other things that is are traditional non biological ingredients – bicarbonate of soda and sal ammoniac, which is ammonium chloride used to make salmakki in Finland. It’s a very unpleasant kind of liquorice.