Groups with a character of large degree

Let $G$ be a finite group of order $n$ and $V$ a simple $\mathbb{C}[G]$-module of dimension $d$. For some nonnegative number $e$, we have $n=d(d+e)$. If $e$ is small, then the character of $V$ has unusually large degree. We fix $e$ and attempt to classify such groups. For $e \leq 3$ we give a complete classification. For any other fixed $e$ we show that there are only finitely many examples.