The definition of richness in food is generally heavy in natural fats in proteins, high in butter and cream of a high quality.

American butter and cream is ludicrously bad. There is just enough taste to it to make the stuff inedible.

Sushi is not a rich food.

Sauces with a lot of butter and fats in them are 'rich', pastries can be rich, depending mainly on the fattiness of the cream and butter.

on edit: French cuisine is generally thought of as one of the richest in the world, hence my use of them as an example.