Something that started getting added to games in the last-generation consoles - hard for me to think of the 360/PS3 era as last-gen since it's the one I'm still comfortably on - were "Achievements". That's what they're called on the 360, I don't know about the PS3.
Each game will have certain things that when you do them, it unlocks a little title or something. It doesn't grant you anything, it's more like a sticker, just verification you did something. It can be beating a boss, or going through a level without killing anyone, or some clever maneuver. Fatal Inertia has one for going over 700 km/hr, The Saboteur had one for kissing 40 ladies (which is a way to evade capture), so on and so forth. It does alter my approach to certain games, because I'll find myself looking over the list of Achievements I haven't unlocked, trying to figure out which ones are doable, and then setting out to complete them. Sometimes it happens, sometimes it doesn't (I haven't reached 700 km/hr). In that sense, it helps extend the play value of the game because it gets me to think about parts of it I might ignore otherwise, or I find myself trying to figure out how to do whatever it is it wants me to do.
On the other hand, I find myself disappointed if I do something in the game, and it doesn't unlock an Achievement. The game designers clearly didn't think that was noteworthy, so I guess it doesn't matter. I don't get obsessed with trying to unlock all the Achievements - I came close on The Saboteur, but on most games I get halfway, roughly - but a part of me likes the little announcement that pops up on the screen when I get one. The mouse hitting the button for the food pellet. Except it's not much of a pellet, really. I'm not sure why it matters. Those games have stored that information somewhere else already, some little notification on the console doesn't make it more real.
But looking back over the achievements helps me remember things I might have forgotten since I last played, or recall things I didn't really notice I was doing at the time. Some trend or pattern in how I approached problems. They kind of work as a map of the path I took through the game, not just marking off the major stuff - killed this Bad Guy, or won some race - but of that tough stretch in Metro 2033 I made it through without dying. Which sparks my memories of that part, with Khan, and the spirits, the strange Anomaly, the horrible part of trying to trigger the explosions to seal off the tunnels before I get ripped to shreds. It's like a little photo album, some of the time anyway.