  Let G be a finite group of order n and V an irreducible representation over
the complex numbers 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<=3 we give a complete
classification. For any other fixed e we show that there are only finitely many
examples.
