Daishi (priest) (大師 (僧))
Daishi is a kind of honorific title in China and in Japan bestowed to priests with high virtue by the Imperial Court, and many of them are okurina (posthumous names).
In China, according to chapter 'Shishigo' (bestowment of title to high virtue monks) in the final volume of "Daiso Soshiryaku" (The Essential History of Great-Sung Monks) by Zanning in the beginning of the Northern Sung Dynasty, the origin is when shigo (title for high virtue priest) 'Sanei Daishi' was Imperially bestowed to Unko of east area of Changan and 'Joko Daishi' was to Sotetsu of west area of Changan in discussion in Engyosetsu (延慶節) on December 10, 870, in the reign of Emperor Iso, in the latter half of the Tang Dynasty. It means that it regards that there were only Hoshigo (kind of title bestowed to monks with high virtue) and Zenjigo (title given to master of Zen Buddhism) existing before that. It states that after that, since bestowment of Daishigo (same meaning as Daishi) and shie (purple robe bestowed Imperially) was issued excessively at the end of the Tang Dynasty, a command to prohibit them was officially announced in 921. It concludes that even at the beginning of the Sung Dynasty, Daishigo was never bestowed until 979.
However, at least in the reign of Shino (King of Jin), an example of Emperor Yodai of Sui Dynasty bestowing the Daishigo 'Chisha' to Zhi-yi when the Emperor had received Bosatsukai (Bodhisattva Precepts) from him can be seen, so actually, the origin of Daishigo can be thought to go back before that time.
In Japan, in August of 866 in the reign of Emperor Seiwa, Daishigo 'Denkyo' and 'Jikaku' were bestowed to Saicho and Ennin respectively, for the first time. As dogo (a pseudonym as a priest) bestowed to priests, there are kokushigo (title given to the most reverend priest), Zenshigo, and so on.
Daishigo in China
Chisha Daishi Zhi-yi (538 - 597): Third founder of Tendai Sect.
Jion Daishi Ki (priest) (632 - 682): Founder of Hosso Sect. Shiso Daishi Chigon (602 - 668): Second founder of Chinese Kegon Sect.
Kajo Daishi Kichizo (549 - 623): Sanron Sect
Daishgo in Japan
Hongaku Daishi Yakushin (827 - 906): Founder of Hirosawa School.
Kogyo Daishi Kakuban (1095 - 1143): Originator of restoration of Shingon Esoteric Buddhism. Established Shingi Shingon Sect.
Enmyo Daishi Mumon Gensen (1323 - 1390): Prince of Emperor Godaigo. Went overseas to Yuan Dynasty.
Josai Daishi Keizan (1268 - 1325): Taiso (great master) of Japanese Soto Sect. Attempted to propagate Soto Sect to general people.
Shinku Daishi Ingen (1592 - 1673): Founder of Japanese Obaku Sect and was from China. Founded Manpuku-ji Temple.
Enko Daishi Genku (1133 - 1212): (Also known as) Honenbo. Founder of Jodo Sect. Originator of Jodo Shinshu (the True Pure Land Sect of Buddhism). After being exiled to Tosa, came back to Kyoto. Endeavored in propagation. Compiled and wrote "Senchaku Hongan Nenbutsu Shu" (the holy writings of the Jodo Sect).
Precise Daishigo were Enko, Tozen, Ejo, Kokaku, Jikyo, Meisho, and Wajun Daishi (7 Daishigo).
Kenshin Daishi Shinran (1173 - 1263): Founder of Jodo Shinshu. After being exiled to Echigo, propagated in Kanto. Compiled and wrote "Ken Jodo Shinjitsu-kyo Gyosho monrui" (Selected passages revealing the true teaching, practice and attainment of the Pure Land), 'Sanjo wasan' (three Buddhist verses), and others.
Yuzu Nenbutsu Sect
Shoo taishi Ryonin (1072 - 1132): Founder of Yuzu Nenbutsu Sect.
Rissho Daishi Nichiren (1222 - 1282): Founder of Nichiren Sect. Wrote "Rissho ankoku ron" (the Treatise for Spreading Peace Throughout the Country by Establishing the True Teaching).