The desert stretches on endlessly.
His story was interminably boring.
The blue sky stretches out endlessly.
とめどなく describes something (often tears, words, or talk) flowing without stopping, whereas 果てしなく is broader and often used for vast spaces or long stretches of time.
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.
Standard kanji-kana spelling; the adverbial form of 果てしない.
Variant using 無 for な; less common in modern writing.
Adverbial form of 果てしない, from 果て (end, limit) + しない (negative of する), literally 'without end'.