Almost all traditional Theatrical Productions have this to an extent—whereas a film or TV series have credits in them, and books feature acknowledgements, etc. etc., if one wants to know who the cast and crew of a particular production are, or sometimes even what the setting is, one needs to have a program from that production (or look it up online...), which will have all that listed. With the exception of Les Misérables, no shows generally use title cards to indicate things, and with the exception of Passing Strange, no show usually has the names of the cast and crew listed aloud at any time during the run.
If you want to completely understand the underlying themes, vague plots, and significance of all - well, most of - the peculiar characters and acts in a given Cirque du Soleil show, you will probably have to buy the souvenir program and/or explore the official website. According to the 20th anniversary book 20 Years Under the Sun, the creators prefer that people watch the more abstract shows (as most of their productions in The '90s were) and create their own interpretations of them rather than have the creators' ideas in mind all along.
In The Drowsy Chaperone, the Man in Chair never names Trix or Geroge's actors, but the CD case to the 2006 recording gives them names. Because this could have been created for the CD alone, it could also be considered Loose Canon.
Likewise, the Man in the Chair is never actually named in the show (justified, because he only ever interacts with one other character). The name "the Man in the Chair" is more a description than a name proper, as it's all really does (besides snark).
In Mrs Hawking, and the semi-steampunk play series by Phoebe Roberts it gives its name to, it's never actually mentioned what Nathaniel does for a living, besides a brief self-identification as a "finance man" in the first sequel Vivat Regina. It takes reading Word of God on the official website to know that he works for his family's venture capital firm.
Almost all of the character's names in Finale are only mentioned in the script, never actually being mentioned in the show.
In Les Misérables, Enjolras' name is only mentioned in the script. However, there is often an improvised "Enjolras!" in most productions of the show.
In Hamilton, the full letter that Hamilton writes John Adams was, for a time, only available in the book "Hamilton: The Revolution". It eventually became a song on The Hamilton Mixtape, called "An Open Letter".