  We prove that for every indecomposable ordinal there exists a (transfinitely
valued) Euclidean domain whose minimal Euclidean norm is of that order type.
Conversely, any such norm must have indecomposable type, and so we completely
characterize the norm complexity of Euclidean domains. Modifying this
construction, we also find a finitely valued Euclidean domain with no
multiplicative integer valued norm.
