One of my favorite things that happens when creating a game is when something I didn't plan in the design happens, or when there is so much going on that I forget what a thing does exactly, and I encounter it anew. In both those cases, a certain level of complexity has been achieved, so that the game develops a life of its own. I think that's one sign of a good game. Of course not every game needs that to happen to be a good game, but for the kind of games I make, and adventure games in particular, and for the kind of feelings I want to evoke when making games, I consider it important and special when it does happen.
Below the complexity threshold, a game can seem dry and predictable. The number of plays which a player feels they can get out of it is low. The stories they tell about the play of the game seem dull. They don't have an 'if only this had happened, or this, or this, then it would have been different.'
So when I make games for my website, Arcane Journeys, I do it in several stages, over a number of years, adding more things as time goes on, and refining the play as well. One time I was testing the game, The Journey to New Earth, and I got caught in a storm on a planet, and heard a strange alien sound nearby. The thing is I did not remember in any way putting this into the game, and the moment was magical for me. I was like, what the heck is this?
Some ways to achieve a complexity threshold are increasing the amount and variety of encounters, or units or situations in your game. In automated systems, adding an AI of some kind to run a simulation in the background really helps.
On computers, achieving this threshold can cause spectacular things to happen. Take Minecraft as an example. But even off the computer there are good examples, like Magic the Gathering, and Chess.
I think the complexity threshold is like a lot of things in game design: you know when you have achieved it, but there isn't an exact formula for making it happen.