Update as of New Year's 2016:
I have been to half of the 40 theatres.
August Wilson Theatre (Jersey Boys)
Booth Theatre (The Glass Menagerie, The Elephant Man)
Broadhurst Theatre (Misery)
Broadway Theatre (Shrek The Musical)
Circle in the Square Theatre (The River, Fun Home)
Cort Theatre (Waiting for Godot, The Cripple of Inishmaan, Fish in the Dark, Sylvia)
Ethel Barrymore Theatre (A Raisin in the Sun, The Curious Incident of the Dog in the Night-Time)
Gerald Schoenfeld Theatre (A Chorus Line)
Gershwin Theatre (Wicked)
Golden Theatre (Driving Miss Daisy, Vanya and Sonia and Masha and Spike)
Lunt-Fontanne Theatre (Beauty and the Beast, The Addams Family)
Lyric Theatre (How the Grinch Stole Christmas, Spider-Man: Turn off the Dark, On the Town)
Majestic Theatre (The Phantom of the Opera)
New Amsterdam Theatre (The Lion King, Mary Poppins)
Palace Theatre (Annie)
Richard Rodgers Theatre (Tarzan)
St. James Theatre (Hair)
Studio 54 (Cabaret)
Vivian Beaumont Theatre (The King and I)
Winter Garden Theatre (Rocky)