Some of the best games are those with strategic elements to them. Where you need to stop and think about how you're going to solve the particulare issue at hand. The best games give you multiple options for a solution. There are definitely times when you really don't want or care to think about your game, but simply to play. Racing games are a good example for those.
But strategy games... I'm talking about things like Hitman, where you need to plan your strategy for infiltrating some location, wiping out your target, and exiting cleanly. There are lots of options and approaches -- snipe the target from a distance? put on a disguise and garrotte him? walk into his location mowing everything down in your path? Options!
Dynasty Warriors is also very good. I've written about it before as a mindless killfest. But between the bouts of mayhem, there is a lot of strategy necessary. Which units do you mow down first? What officers do you go after? Can you wipe out the guard retinue before they escort that officer off the map? When the map first starts up, it takes some sitting back and some thought to plan an approach to the mission.
One of my favorite games when I'm in a strategy mood is Ghost Recon. It takes some serious thought to figure out how to navigate the map and take out baddies without having them kick your team's ass. With Ghost Recon, I really want to be able to complete the level without damage to my team. Thus, planning an approach to the level is quite important. Patience is also helpful with this game. Tracking down the baddies, waiting for them to hit that sweet target spot, or creeping along to ambush them.
Ghost Recon is simply a great game for whackin' bad guys, and doing it in style.