To what extent is the circulating story true about a man from the Salaf who said, "There is no god but Allah, as many as there were, as many as there will be, and as many as the movements and stillness," and after a year, the angels said, "We have not finished recording the good deeds of the past year"?

It is not permissible to attribute reward to any supplication without authentic evidence from the Qur’an and Sunnah (definition). The tradition mentioned in the question has not been reported in the books of the scholars of hadith (definition), so it is not permissible to disseminate it except as a warning against it. One must be cautious about spreading falsehoods about the religion. Sheikh Ibn ‘Uthaymeen described the supplication "La ilaha illa Allah, as many times as there has been and will be…" as akin to exaggeration, preferring what has been authentically reported in the Sunnah, such as "Subhan Allah wa bihamdihi, عدد خلقه، ورضا نفسه، وزنة عرشه، ومداد كلماته" (Glorified is Allah and praised be He, as many times as His creations, as pleases Him, as much as the weight of His Throne, and as much as the ink of His words) as it encompasses all glorification.