In the effective topos there exists a chain-complete distributive lattice with a monotone and progressive endomap which does not have a fixed point. Consequently, the Bourbaki-Witt theorem and Tarski's fixed-point theorem for chain-complete lattices do not have constructive (topos-valid) proofs.
['Andrej Bauer']
On the Failure of Fixed-Point Theorems for Chain-complete Lattices in the Effective Topos