I have googled for a little while and found pages and pages of rules and explanations but no documentation. Are there any documented games where the moves have been recorded? It's so ridiculous, I don't believe that anyone has ever been able to finish a game, let alone remember all of the piece movements.
The answer is yes.
According to jawp article, a game of Taikyoku Shogi was played for Japanese TV show "Tribia-no-Izumi" (=Fountain of trivia), broadcast on 19th May 2004. The game continued 32 hours 41 minutes and 3805 moves before the first player won. This seems to be the first full-recorded game of the variant.
Also, there is a webpage dedicated to Taikyoku Shogi with a playing webservice（大局FLASH, requiring Flash Player）.