Kaplansky's second conjecture (on Hopf algebras) deals with "*admissible*" coalgebras: He calls a coalgebra admissible, if there is an algebra structure making it a Hopf algebra. The conjecture states that: 
>a coalgebra $C$ is admissible if and only if any finite subset of $C$ lies in a finite dimensional admissible subcoalgebra. 

As far as I know there are counterexamples refuting the conjecture. (the first one was stated by Larson if I remember correctly). 

However, my question has to do with whether there is some similar conjecture (or some result) for what could be called an **admissible algebra** : Given an algebra $A$, under which conditions can a coalgebra structure be found on $A$ such that it will be turned into a Hopf algebra? Furthermore, when would such a coalgebra structure be unique? 

**Edit:**  What could be an example of a **non-admissible algebra**, in the sense of the above definition?