A huge part of the enjoyment of playing a video game is the experience of breezing your way through the campaign and just enjoying the story. However, many gamers that I've known over the years simply don't take advantage of difficulty levels, and are satisfied with beating the game once. Go ahead and play some multiplayer if the game offers it, but I dare you to complete even one of these games on the hardest difficulty.
One extremely challenging game is God of War III. This God of War series follows the story of Kratos and his quest to take down the Greek gods of Olympus. Many view this game as the best in the series, and also the most difficult. You shouldn't struggle too much on your first playthrough, but on 'Chaos' mode (the hardest difficulty level) you'll want to throw your controller out the window.
Enemies attack without mercy, recover immediately, and do insane amounts of damage! In my opinion, one of the hardest fights in the game is one where you defend your family from endless waves of Kratos clones. It just keeps going and going without mercy. In addition, the already challenging boss fights become even harder, so get ready for a long, frustrating challenge.