Take the turn based tactical game genre. As I've mentioned, I was playing La Pucelle. This game suffers in this respect because the player isn't actually doing things all the time. For example, after setting up a miracle, I would often set down the controller and browse gamedev for a while as the animation for the dark portal destruction takes place. Fortunately, the creators of La Pucelle also supply a response to this problem in the form of their follow-up game, Disgaea. In Disgaea, not only does the animation for the destruction of panels accelerate after you trigger a geo-panel chain, but you feel more need to pay attention to the animation since you can be harmed in the resulting chaos.
Another example of a game getting it (mostly) right is Tales of Symphonia for the GameCube. This game demands near constant player interaction, particularly in how the combat system runs in real time. The only problem I see it suffering from in terms of player engagement is the skit system, where there just doesn't seem to be anyway to accelerate the text displayed. It's also relatively easy to break out of a skit by accident. Since quite a bit of the character development and a few story hints happen in the skits, this become annyoing.
Another example of player engagement is from sprinkling little bonuses in the game. I don't mean easter eggs, though those are nice. Things like Tuckerizing (giving minor elements names that the reader/player might be familiar with, such as naming the moon's capital Heilein) or references to popular culture/parody. A good example of this was Fallout, where you'd be on the look-out for the next Monty Python reference or what not. A not-so-good example of this was Fallout 2 where they just went overboard and those kinds of things came up so often that it didn't feel like it was an actual reward for the attentive player.
The basic rules seem simple, punish the player whenever they take their hands off the controls or their eyes off the screen. Reward the player for paying attention. Even for non-interactive applications, tiny bonuses for someone paying attention can increase engagement by a large degree. The classic example of that would be the Pipes Windows Screensaver (TEAPOT!).