In fact, there's a very important class of representations which are typically not faithful (non-exact, in your terminology): characters.
A character of a group $G$ is a homomorphism $ \chi:G\longrightarrow{\Bbb C}^\times. $ It corresponds to the case where $\dim V=1$.
The set $\hat G$ of caracters of $G$ is itself a group: you can multiply characters by $(\chi_1\cdot\chi_2)(g)=\chi_1(g)\chi_2(g)$
Some groups have very few characters. For instance, the symmetric group ${\cal S}_n$ has only two characters, the constant map and the sign map $ {\rm sgn}:{\cal S}_n\longrightarrow\{\pm1\} $ giving the sign of a permutation. The alternating group $A_n$ with $n\geq5$ has no non-trivial characters, as does every group lacking non-trivial normal subgroups.
On the other hand, if $G$ is abelian there are usually lots of characters. For instance, it s not too hard exercise to prove that if $G$ is finite, then ${\hat G}\simeq G$ (although non canonically).