どこまでも conveys 'all the way, however far', giving a sense of persistence or extension in any direction, while 果てしなく emphasizes that no end is perceivable at all.
Written forms
果てしなく
Standard kanji-kana spelling; the adverbial form of 果てしない.
果てし無く
Variant using 無 for な; less common in modern writing.
Etymology
Adverbial form of 果てしない, from 果て (end, limit) + しない (negative of する), literally 'without end'.
