Long vocalic r occupies the eighth place in the varnamala and is one of the clearest illustrations of what kind of thing the Sanskrit alphabet is. Every other simple vowel in the language comes in a short form and a long form, and the pairing is systematic: a with long a, i with long i, u with long u. Vocalic r is a full vowel by the grammarians' own analysis, carrying a syllable with no other vowel beside it, so the pattern requires that it have a long partner, and the table duly provides one.
The language then almost never uses it. Where the sound does occur it is produced by grammar rather than drawn from vocabulary. Its regular home is the genitive plural of nouns whose stems end in r, a class that includes the words for father, mother, brother and several agent nouns, where the ending lengthens the stem vowel; a handful of derived and compounded forms account for the rest. There is no common word in which it appears as an ordinary part of the root. Its vowel sign is the curl written beneath the consonant for short vocalic r, with an additional stroke.
The contrast with an alphabet like the Greek or the Hebrew is instructive. Those are inherited sequences of signs, each of which exists because scribes needed it, and their order preserves the order of the Phoenician series they descend from. The varnamala is not that. It is a phonetic table produced by analysis, laid out by place of articulation and by length, and it contains places that the analysis demands whether or not the language fills them. Long vocalic r is the first of those places, and the two letters that follow it are the emptier still.