Possibility of π–π stacking in non-aromatic systems
There is a paper titled "Amide-pi interactions between formamide and benzene",  doi: 10.1002/jcc.21212. It has been cited by a paper on a crystal structure on a urea transporter (the interaction is between a peptide bond of a ligand and aromatic side chains of the protein).

Technically, the interaction between the arginine side chain and aromatic groups could also fall under your heading, but it is usually described as a cation pi interaction.