Not sure exactly what you mean here, but maybe an example would be a relation with 6 attributes A, B, C, D, E, F with dependencies

A -> {B,C,D,E,F} (so {A} is a candidate key)

{B,C} -> D

{C,D} -> E

F -> {B,D}

so that there is a transitive dependency {B,C} -> {C,D} -> E which involves no prime attributes. Such a relation is in 1NF but not in 2NF or any higher normal form, and looking at possible update, delete, and insert anomalies in such a relation usually provides a pretty good indication of why not being in 2NF is a terrible idea.

Obviously there are simpler examples of that sort, for example a relation with four attributes {A,B,C,D} and dependencies

A -> B ({A} is the primary key)

B -> C

C -> D

with transient dependency B -> C -> D involving no prime attribute, but these are a bit boring and maybe too simple to be realistic.