What is the authenticity of the remembrances mentioned in the book, such as "Allah bears witness that there is no god but He..." 300 times, and "And whatever good you put forth for yourselves..." 100 times, for increasing good deeds? And are there any remembrances that are recited more than 100 times?

Limiting ذكر (remembrance of God) to a specific number not prescribed by the Lawgiver is an innovation (bid‘ah). Therefore, reciting the verse: {Allah bears witness that there is no deity except Him, and [so do] the angels and those of knowledge – [He is] maintaining [creation] in justice. There is no deity except Him, the Exalted in Might, the Wise} 300 times, and the verse: {And whatever good you put forward for yourselves} 100 times, has no basis in Islamic Law. One must be cautious of books of adhkar that do not adhere to the Sunnah. Books such as "Fortress of the Muslim" (حصن المسلم), "Al-Adhkar" by An-Nawawi, "Al-Kalim At-Tayyib" by Ibn Taymiyyah, and "Al-Wabil As-Sayyib" by Ibn Al-Qayyim are recommended.