Let $G$ be a connected linear algebraic group. This question concerns Hilbert's 14th Problem for the adjoint action of $G$ on itself. Let $k[G]^G$ denote the algebra of regular functions on $G$ invariant under the conjugation action. It is well-known that if $G$ is reductive, then $k[G]^G$ is finitely generated and that it is in fact a polynomial algebra on $r$ generators, where $r$ is the reductive rank of $G$. The main question concerns non-reductive groups:
Question: Is $k[G]^G$ always finitely generated? If not, can we give a necessary/sufficient criteria for when it is finitely generated? Can one "describe" $k[G]^G$ even if it is not finitely generated?
For instance, what does the invariant ring for the group $U_n$ of $n\times n$ unipotent upper triangular matrices look like?