Which collaborative team has written the most number of musical theatre shows (e.g. Puccini/Fontana, Gilbert/Sullivan, Gershwin/Gershwin, Hart/Rodgers, Rodgers/Hammerstein, Rice/Lloyd Webber, etc., etc...)?

I would guess Richard Rodgers and Lorenz Hart (at 28) but I'm not at all sure how one might confirm that as "most."

Among famous musical theatre duos, Gilbert and Sullivan are widely seen as the most prolific when writing full-length works that are still regularly performed. They teamed up on 14 comic operas, including hits like The Mikado and H.M.S. Pinafore.

Other teams, like Rodgers & Hart or Gershwin & Gershwin, may have written more shows numerically, but many of theirs were shorter or less enduring. Teams like Rodgers & Hammerstein or Rice & Lloyd Webber produced fewer collaborations overall, though their impact was massive.

In short, Gilbert and Sullivan win for sheer consistency in full-scale theatre that has stood the test of time.
https://saraleeetter.com/2016/06/29/gilbert-sullivan-101-all-fourteen-operas/