LSJ
α, ον, Apeiraean, and Ἀπείρηθεν, Adv.
from Apeire, both in Od. 7.8, 9 γρῆϋς Ἀπειραίη…, τήν ποτ’ Ἀπείρηθεν νέες ἤγαγον. Apeire seems to be Limitless-land (from ἄπειρος²), an imaginary place; Hsch. expl. by ἠπειρωτική.
Liddell-Scott-Jones, Greek-English Lexicon (9th ed., 1940)