#JavaScript, 48 bytes

Input and output as strings. Returns the `nth` term of the sequence.

    f=s=>s==(s=s.replace(/(.)\1/g,x=>x[0]*2))?s:f(s)

[Try it online](https://tio.run/##TYvfDoIgHEbveQrHFbRCfpgO2rAHMS@YqauZOmnNtyfwT@u7O2fne5qPsdX0GN@nfrjXzjXa6txqTay2bKrHzlQ1iQmjN4jb46zzueDlQVB6tZeGWOqqobdDV7NuaEmBMMfHaFkcRxxh@YcSYSHEKjyeBcKQJckiAntMU1CBPQoOyhulpNyNhAxhFbYZkCBDtFchgqUC7gerAg7bTeHfDSQq2cuMpKHUfQE)