There's nothing in the Rush catalog that's been played live more than the first two parts of their 1976 masterpiece "2112." They've been the highlight of nearly every Rush show since the album came out, surfacing more than 1,300 times onstage. Even on the 2012/2013 Clockworks Angels tour where there wasn't a single 1970s song in the main set, they still made sure to wrap up most nights with the opening segments of "2112." Doing otherwise would probably caused an audience revolt. When their Rock and Roll Hall of Fame induction set was limited to three songs, they made sure one of them was the opening of "2112."
But the song is divided into seven segments that take up the entire first side of the vinyl record. They've condensed it a bunch of different ways over the years, but even on the original 2112 tour in 1976 they'd leave out segments like "Part V: Oracle: The Dream" to save time. The song is simply too long. The only exception came on the 1996/97 Test for Echo tour. The first set of every show that tour wrapped up with the complete "2112" suite, all 21 minutes of the thing. For Rush fans, it was a dream come true. Here's pro-shot video of them doing it at Toronto's Molson Ampitheater on June 30th, 1997. It was one of the final stops of the long tour.
Nobody knew it at the time, but the end of Test for Echo tour marked the beginning of a very difficult time for Rush. Less than two months after it wrapped, drummer Neil Peart's 19-year-old daughter Selena, his only child, died in a car accident. The following summer, his wife Jacqueline died of cancer. The unbearable tragedies caused Peart to put his drums aside and go on a lengthy, cross-continent motorcycle journey that he eventually chronicled in his book Ghost Rider. Nobody knew if Rush would ever play again, but by 2002 he was finally able to return to music.
The group spent much of the next thirteen years touring and recording, but in 2015 Peart announced that his body simply couldn't handle the rigors of playing live. They said farewell with a brief 2015 tour and have been completely inactive since. There are rumors that Geddy Lee and Alex Lifeson might do some sort of project with another drummer, using a name other than Rush, but they haven't said a single public word about that. It could very well be little more than speculation and fans trying to wish a two-thirds Rush tour into existence.