TITLE: Properly "transfinitely" Euclidean domains
QUESTION [6 upvotes]: Are there integral domains which admit ordinal-valued Euclidean functions but not $\mathbb{N}$-valued Euclidean functions?

REPLY [6 votes]: Yes, they exist. Even if the problem is left open in the the papers of T. Motzkin and P. Samuel cited in Comparing different Euclidean algorithms on a Euclidean domain 
the problem is solved in 
J. Hiblot, Des Anneaux euclidiens dont le plus petit algorithms n'est pas  valeurs finies, C. R. Acad. Sci. Paris, 281 (1975), 411-414 with correction in vol 287.
M. Nagata, On Euclid algorithm. C. P. Ramanujama tribute, pp. 175 Tata Inst. Fund. Res. Studies in Math., 8, Springer, Berlin-New York, 1978.
See also: 
Pag. 195 in P. M. Cohn, Free Ideal Rings and Localization in General Rings
Rod Downey, Euclidean Domains and Euclidean Functions, http://homepages.mcs.vuw.ac.nz/~downey/ntu_13.pdf
Pete L. Clark, A note on euclidean order types, http://arxiv.org/abs/1208.0977