The intermediate Hamiltonian (IH) coupled-cluster method makes possible the use of very large model spaces in coupled-cluster calculations without running into intruder states. This is achieved at the cost of approximating some of the IH matrix elements, which are not taken at their rigorous effective Hamiltonian (EH) value. The extrapolated intermediate Hamiltonian (XIH) approach proposed here uses a parametrized IH and extrapolates it to the full EH, with model spaces larger by several orders of magnitude than those possible in EH coupled-cluster methods. The flexibility and resistance to intruders of the IH approach are thus combined with the accuracy of full EH. Various extrapolation schemes are described. A pilot application to the electron affinities (EAs) of alkali atoms is presented, where converged EH results are obtained by XIH for model spaces of approximately 20,000 determinants; direct EH calculations converge only for a one-dimensional model space. Including quantum electrodynamic effects, the average XIH error for the EAs is 0.6 meV and the largest error is 1.6 meV. A new reference estimate for the EA of Fr is proposed at 486+/-2 meV.