The main purpose of this paper is to construct ergodic, nonsingular, conservative n-to-one endomorphisms which preserve no equivalent sfinite measure. While such examples are known to exist w6, 9, 12, 19x, our examples are fundamentally different from the existing ones. It was noticed independently in w6, 19x that the two-sided type III Bernoulli shift of Hamachi w9x is the natural extension of an exact two-to-one endomorphism with no equivalent s-finite invariant measure. Also in w12x it is shown that Cartesian products of finite measure-preserving exact with type III automorphisms yield examples of n-to-one endomorphisms with no equivalent invariant measure. In Section 5 we give an example of a two-to-one endomorphism whose ergodic nonsingular measure is neither exact nor is it equivalent to a product measure of an automorphism with an exact endomorphism.