One game with a version of this mechanic is Fabled Fruit. It's a worker placement/ hand building game. There are 6-10 possible actions you can take each turn: the initial 6 are relatively simple but as the game goes on, old actions get discarded and new ones, often with more complex mechanics, take their place. You're encouraged to keep the game in the new more complex state the next time you play it.

Even the initial state is too complex for young children but it might suit what you're looking for if that's not actually a requirement - I don't think you'd really enjoy playing phase 1 of his Carcassonne adaptation either.