For a large class of Hardy fields their extensions containing non--germs are constructed. Hardy fields composed of only non--germs, apart from constants, are also considered.
@article{bwmeta1.element.bwnjournal-article-doi-10_4064-ap93-3-8, author = {Krzysztof Grelowski}, title = {Extending Hardy fields by non-$^{$\infty$}$-germs}, journal = {Annales Polonici Mathematici}, volume = {93}, year = {2008}, pages = {281-297}, zbl = {1147.26018}, language = {en}, url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-ap93-3-8} }
Krzysztof Grelowski. Extending Hardy fields by non-$^{∞}$-germs. Annales Polonici Mathematici, Tome 93 (2008) pp. 281-297. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-ap93-3-8/