It's not one of Shakespeare's popular plays. I don't think I'd read or seen a production until I started this list. I'm rather surprised that it isn't more popular. For one thing, it isn't quite as dismal and unending as the War of the Roses plays.
It also has a great mystery, Arthur--the rival to King John's accession to the English throne--disappeared when he was fifteen to sixteen. Like Richard III's later nephews, he was likely murdered; it was apparently quite easy in the Middle Ages to kill and hide the bodies of teenage boys.
Philip of Cognac as "Philip the Bastard," the son of Lady Faulconbridge and King Richard, is unique in that he is a main character who acts beside the nobility and also produces much of the play's dark humor. Despite his self-serving nature, he has a grasp of basic morality and operates as a kind of Loki commentator. Unlike Edmund from King Lear, he is not (necessarily) villainous.
For the purposes of this list, the play doesn't produce a grand passion, but Shakespeare uses a marriage--Blanche of Castile to the Dauphin--to temporarily establish peace between England and France. Shakespeare witnessed the ongoing uncertainty about who would inherit the English throne after Henry VIII (he was 17 when Henry VIII died) and after Elizabeth (he lived into King James I's reign). The world in which he operated discussed royal marriages as much as modern people do, with better reason.
Shakespeare's approach here--marriage as a form of peace--is in keeping with the beliefs of those royal families. That is, it was "given" that a marriage WOULD influence political outcomes, even if not the exact political outcomes that the marriage was expected to create.
So Blanche, played by Jennifer Mogbock, in King John is placed--like the teen Arthur--in a kind of custody battle with her marriage when England and France (pressured by the Vatican) part ways again. Like Eleanor of Aquitaine's marriage to Henry II, every choice regarding the marriage will have ramifications. (After all, would any other set of parents have produced such ambitious, intelligent, charismatic, and reckless sons?)
Ultimately, Blanche survived, became queen of France and exercised a great amount of power!