The King of Siam has several wives and lots and lots of children.
There is a very sensual scene where as the king and Anna are about to dance she slowly walks towards her and places his hand around her waist and pulls her closer to him. She is wearing a sleeveless dress and submits to him with her hands behind her back.
Several women lift their gowns as they flee from a room and unintentionally flash the English ambassadors (it is implied earlier that they aren't wearing proper undergarments), but no nudity is seen.
Although no sex is ever shown, it is strongly implied that Tuptim and Lun Tha are secret lovers.