Some notes on Qiaochu's answer:
We can sometimes compute the decomposition into irreducibles of $V \otimes V^*$ using character theory. The number of irreducibles in $V \otimes V^*$ (or rather the squared sum over isomorphism classes) is of course the second moment of the absolute value of the character of $V \otimes V^*$, which is the fourth moment of the absolute value of the character of $V$. If this is $<5$, we know exactly how to decompose it into irreducible representations.
The same idea works for semisimple Hopf algebras. By the Artin-Wedderburn theorem, a semisimple Hopf algebra, as a diagonal representation of itself, is equal to a sum of the tensor product (over the automorphism algebra) of each simple representation with its dual.