Are there integral domains which admit ordinal-valued Euclidean functions but not $\mathbb{N}$-valued Euclidean functions?



Properly "transfinitely" Euclidean domains
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](https://mathoverflow.net/questions/143065/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>


