Surat al-Tawba is the only surah in the Quran that does not open with the Basmala (In the Name of Allah, the Most Compassionate, Most Merciful). Two explanations are related from the sahaba. Uthman ibn Affan RA, who compiled the mus'haf, was asked why he placed al-Tawba directly after Surat al-Anfal; he explained that al-Tawba was one of the last surahs revealed, that the sahaba were unsure where it belonged, and that they placed it after al-Anfal because the subject matter (warfare) was similar, and, being uncertain whether the two surahs were truly connected, they did not insert a Basmala between them. Some of the tabi'un, on this basis, counted Anfal and Tawba as a single surah (making 113 surahs in total), though the position of the wider ummah, and the later ijma, is that they are two separate surahs. Separately, Ali ibn Abi Talib RA was asked by one of his sons why there is no Basmala in Surat al-Tawba; he explained that the surah is where Allah cuts off His ties with the pagans, and that it is not befitting to begin such a surah with an invocation of His mercy; the surah's opening word, bara'ah (disavowal), itself signals this. Retelling this explanation in a later lecture, Sh. Yasir Qadhi glosses Ali RA's reasoning further: the Basmala is what one writes at the beginning of a contract of protection (an invocation of rahma (mercy)) whereas Surat al-Tawba is a surah of bara'ah (complete cutting-off and dissociation), so it would not be fitting to open a surah of dissociation with an invocation of mercy.
The absence of the Basmala also raises the separate question of who ordered the surahs within the mus'haf. Sh. Yasir Qadhi notes that, unlike the ordering of the ayat within each surah (agreed by all to have been fixed by the Prophet ﷺ under revelation), the ordering of the surahs themselves appears, on the evidence of Uthman's report and of sahaba who had arranged their personal copies differently before standardization, to have been the sahaba's own doing, which became binding on the ummah only once Uthman's compiled mus'haf achieved ijma.
O Amir al-Mu'minin, why did you put Surat al-Tawba after Surat al-Anfal?
Al-Tawba was one of the last surahs revealed, and the matter was unclear to us where it should go. Therefore, we put it with Anfal because the content is the same. And we did not put a Basmala not knowing if the two Surahs are connected or not.
Why is there no Basmala in Surat al-Tawba?
This is a Surah where Allah cuts off His ties with the pagans; so it is not befitting that He begins it with His mercy.