Cole Porter potentially raises the bar, being that in the middle of his career, he was often writing two musicals in a year, but most of these shows didn't run for more than a few months, often closing before the next one opened, and only three of them (Anything Goes, Kiss Me Kate, and Can Can) have every been revived on Broadway.
Rodgers and Hart do better, having often written three or more shows in a year. They had four shows open on Broadway in 1926, but two of them closed before the other two opened -- opening consecutively on December 27th and 28th. Them having three shows running at once sounds plausible at the rate they wrote, but if you're writing three shows a year, how many of those can you expect to be hits?
But that was all pre-Hammerstein. Once Oklahoma! entered the picture, musicals stopped being mass-produced for the sake of hit songs, resulting in composers producing fewer shows, but more potential long-running hits. I don't know if anyone's ever had more than three shows running on Broadway at once, even posthumously, but if so, someone like Richard Rodgers or Cole Porter, writing for pre-Hammerstein Broadway seems like the most likely. In revivals, Rodgers and Hammerstein have an advantage in that they wrote five big popular hits that have continued to remain popular, which is more than most people. Among most major Broadway composers and lyricists, two or three big hits -- sustainable big hits -- seems more like the norm.
And now a follow-up. Who is the individual who has been connected to the most shows running on Broadway currently? Or at one time in general? A lighting designer? A violinist? An ensemble member? It's possible that there's some actor or actress who was in the ensemble of Phantom when it first opened, and then moved to the ensemble of Chicago, and then Lion King, and so on.
Upon review, Andrew Lloyd Webber seems to have done this before. The last revival of Jesus Christ Superstar slightly overlapped with the subsequent revival of Evita, and all this while Phantom was still running. This being the advantage of having a show run for twenty plus years.
Speaking of which, John Kander has had three shows on Broadway at once, though a little more loosely. He wrote the dance arrangements for Gypsy, the 2008 revival of which overlapped with the original production of Curtains, while Chicago was (and is) still going on as a long-running show.
On a whim, I looked at Boublil and Schonberg, since Miss Saigon and Les Miserables both had long overlapping runs. But, surprisingly, Martin Guerre was never on Broadway, and their next show, The Pirate Queen, has after Les Mis and Miss Saigon had both closed.
But on the subject of Les Mis, Miss Saigon, Cats, and Phantom, as a producer, Cameron Macintosh might be in the lead, having been behind several long-running shows. And sure enough, from April 11th, 1991 (opening of Miss Saigon) to September 10th, 2000 (closing of Cats) those four Cameron Macintosh productions were running on Broadway. And in that nine-year stretch, he also had other productions with shorter runs, none of which overlapped. So at some points, Cameron Macintosh had five productions running on Broadway at once. How's that for impressive?