I know overmorrow (the day after tomorrow) and ereyesterday (the day before yesterday) themselves are obsolete alike. I would like to know whether English has ever had words for one day farther than that, I mean "the day after overmorrow" and "the day before ereyesterday".
There is an historical record of third morrow meaning the same thing as overmorrow. See OED sense 2b of morrow.
So presumably there is no reason one could not use "fourth", "fifth" or "sixth" morrow.
2b. the third morrow: the next day but one. the next morrow: the day after. Frequently used adverbially. Now rare (arch. and literary). c1325 in G. L. Brook Harley Lyrics (1968) 57 Hire blisse sprong þe þridde morewe. ▸ a1393 Gower Confessio Amantis (Fairf.) ii. 2713 (MED) Wherof his lord..A seknesse er the thridde morwe Conceived hath of dedly sorwe. a1400 (▸a1325) Cursor Mundi (Vesp.) 24549 (MED) He hight to rise þe thrid moru. a1450 Generides (Pierpont Morgan) (1865) 5621 On the thrid morow Bellins, the king, Rose erlie or the day gan spring. a1522 G. Douglas tr. Virgil Æneid (1957) iii. ii. 97 Our navy sall, with help of Jupiter, The thrid morow be at the cost of Crete. 1533 J. Bellenden tr. Livy Hist. Rome (1901) I. 65 The nixt morow he wald mak ane sacrifice lustrale als sone as licht apperit. a1586 Sir P. Sidney Arcadia (1590) ii. 162 He would the third morrow after, before the walles of the towne strike off Antiphilus head. a1645 W. Browne tr. M. Le Roy Hist. Polexander (1647) i. iii. 65 The Assembly..thought it fit to resolve of nothing that day, but that they should meet againe the next morrow to conclude all things. a1722 J. Lauder Hist. Notices Sc. Affairs (1848) II. 651 The Privy Counsell ordained the Criminal Court to sit on him the nixt morrow. 1893 A. Webster Portraits 20 The next morrow they will feel their ease And sigh with sleek content, or laugh elate. 1906 C. M. Doughty Dawn in Brit. I. iii. 157 They, with Arunt, Will the third morrow march, in aid, towards Clusium.