It's actually interesting, but from those um... leaked papers... we learned that Gaga's Interscope contract has restrictions on number of songs on an album and even song length, plus who she works with production-wise has to be approved by the label before it's even in the recording stage. So there are actually very few, if any, unused songs that ever get finished for her. The label also has full discretion to choose which songs are singles and get music videos. I wonder if Madonna's is similar in a way.
Taylor's Republic Records deal affords her almost unparalleled creative freedom for a megastar artist. She not only owns her own masters now (so she can record whatever and whenever she likes even if it never gets released), but she can also work with whoever she wants to and without informing anyone. The label is basically only for distribution and marketing now, so her own Taylor Nation in-house team basically did all the graphics and lyric videos until it was time to send everything to be shipped out for production and distribution for release.
Ariana also seems to get similar freedom on Republic and seems to be recording a lot and has been able to spontaneously release a lot more songs in the streaming era than we used to be able to get, but she probably makes less on her music because she shifts fewer physical sales and not all of her recordings make it to physical release.
Now we know Madonna has a lot of demos and other recordings that never make it to market, so obviously her recording flexibility probably has more freedom than Gaga's, but it's interesting to wonder what other restrictions she has to deal with.