We study exact solutions to cosmological topologically massive gravity coupled to topologically massive electrodynamics at special values of the coupling constants. For the particular case of the so-called chiral point l mu(G)=1, vacuum solutions (with vanishing gauge field) are exhibited. These correspond to a one-parameter deformation of general relativity solutions, and are continuously connected to the extremal Banados-Teitelboim-Zanelli black hole with bare constants J=-lM. At the chiral point this extremal Banados-Teitelboim-Zanelli black hole turns out to be massless, and thus it can be regarded as a kind of ground state. Although the solution is not asymptotically AdS(3) in the sense of Brown-Henneaux boundary conditions, it does obey the weakened asymptotic recently proposed by Grumiller and Johansson. Consequently, we discuss the holographic computation of the conserved charges in terms of the stress tensor in the boundary. For the case where the coupling constants satisfy the relation l mu(G)=1+2l mu(E), electrically charged analogues to these solutions exist. These solutions are asymptotically AdS(3) in the strongest sense, and correspond to a logarithmic branch of self-dual solutions previously discussed in the literature.