Abstract: We show that the translation operator Tf(z) --> f(z+1), acting on certain Hilbert spaces consisting of entire functions of slow growth, is hypercyclic in thesense that for some function f in the space, the orbit {T^n f: n >=0} is dense. We further show that the operator T-I can be made compact, with approximation numbers decreasing as quickly as desired, simply by choosing the underlying Hilbert space to be sufficiently small. This shows that hypercyclic operators can arise as perturbations of the identity by ``arbitrarily compact'' operators. Our work extends that of G.D. Birkhoff (1929), who showed that T is hypercyclic on the Fréchet space of all entire functions, and it complements recent work of Herrero and Wang, who were the first to discover that perturbations of the identity by compacts could be hypercyclic.