The root √AR has a long and complex history in Tolkien’s writing. For many years, it was the basis for the word ar “and”. Its earliest precursor was the root ᴱ√ARA or ᴱ√ƷARA in the Qenya Lexicon variously glossed “spread, extend sideways” or “wide places” (QL/32). The Gnomish derivatives of this root such G. garw “sown field” (GL/38) vs. ᴱQ. arwa made it clear the true primitive form was √ƷARA (in Gnomish, ʒ- > g-). Some of the early derivatives of √ƷARA such as G. gar(th), ᴱQ. arda “place” were later transferred to the root ᴹ√GAR so they could retain this gar-/ar- distinction.
Of the derivations that remained under √AR, the most notable were ᴱQ. are “beside, along” and the conjunction ᴱQ. ar(a) “but” (QL/32). The latter changed in meaning to ar “and” by the end of the 1920s, for example in the Oilima Markirya poem. This carried into the 1930s paradigm for the root ᴹ√AR, as seen by its entry in The Etymologies with its derivatives ᴹQ. ara “outside, beside” (the basic sense of the root) and ᴹQ. ar “and” (Ety/AR²). The most common Noldorin word for “and” in this period was likewise ar (TAI/150; SD/128-129), and in prefixal form ar- “outside, beside” sometimes developed a privative sense “without”, most notably in arnediad (†arnoediad) “without reckoning, numberless” as in N. Nirnaith Arnediad “(Battle of) Unnumbered Tears” (Ety/AR², NOT) which in Sindarin became Nirnaeth Arnoediad.
In some etymological notes from the 1950s Tolkien retained the root form ara “alongside” (VT43/33), but there were already cracks forming in this system, forced by Tolkien’s decision that the Sindarin word for “and” was a rather than ar, a change that first appeared Lord of the Rings drafts from the 1940s (TI/182). By the late 1950s Tolkien was experimenting with new roots √AD(A) and √AS for the meaning “beside” and the derivation of Q. ar, S. a “and”; see those entries for later developments in this semantic space of “beside”.
As for the root √AR itself, it shifted in meaning to “beyond, further than” in Quenya Notes (QN) from 1957, becoming the basis for “royal” roots like √ARAN “king” or √ARAT “noble” (PE17/147). In this revised meaning, it might still be able to retain a “privative” sense in Sindarin words like †arnoediad “unnumbered” (perhaps = “✱beyond numbering”), though it is also possible Tolkien simply never revisited the etymology of this Sindarin word.