Please remember: none of videos and books can replace a practice in dôjô!
tai (太) - big, great
kyoku (太極) - 1) end (substantive), pole (substantive); 2) extreme (adjective); 3) "go to the end" (verb), "study thoroughly" (verb), "carry to extremes" (verb).
There exist six Taikyoku katas: Taikyoku Shodan, Taikyoku Nidan, Taikyoku Sandan, Taikyoku Yondan, Taikyoku Godan, Taikyoku Rokudan.
Taikyoku Shodan, Taikyoku Nidan and Taikyoku Sandan were most probably the result of the work of the following karateka: Funakoshi Gichin, Funakoshi Yoshitaka, Egami Shigeru, Hironishi Genshin, Kugimiya Y. (I am still trying to find the first name!). These kata (together with Ten no kata) were created in the late 1930s and early 1940s.
Taikyoku Yondan, Taikyoku Godan and Taikyoku Rokudan were created by the pioneer of European karate Henry Plée. The book by H. D. Plée "Karate: beginner to black belt", published in 1967, is the only book, currently known to me, where these kata are shown. (A note to my students: I got a copy of this book in my library, so ask me if you are interested to read it.)
Taikyoku Shodan, Taikyoku Nidan and Taikyoku Sandan are shown in:
Gichin Funakoshi "Karate-Dô Kyohan. The Master Text", 2nd edition, 1958. (Note: a number of the edition is critical, as the first edition, published in 1935, does not contain Taikyoku kata.)
Besides, Taikyoku Shodan can be found in some other books, for example in:
Fiore Tartaglia "Shôtôkan-Kata bis zum Schwarzgurt"
H. D. Plée "Karate: Beginner to Black Belt" (WARNING!!! Taikyoku Nidan in this book is shown with chûdan oi zuki instead of jôdan oi zuki as techniques 2, 4, 10, 12)
Taikyoku Yondan, Taikyoku Godan and Taikyoku Rokudan are found in:
H. D. Plée "Karate: Beginner to Black Belt" (currently I don't know any other source for these three kata)
Taikyoku Sandan IMPORTANT!!! For some reason, after gedan barai, the performer is doing chûdan oi zuki instead of jôdan oi zuki!!!
The following three videos show the kata precisely according to the book by Henry Plée: