I've been doing a lot of two things as of late, playing Bioware RPG's and thinking. If one is to do this too much, they start to realize some truly odd things about the way the story in these games (namely the Mass Effect series and Dragon Age: Origins) works as far as how much effect the character you play (you referring to the person playing the game) has on the story as a whole, or at least the small details of it. Let me elaborate.  

Let's say i'm playing Mass Effect 2, as good 'ol Commander Shepard. I'm running around on the Citadel, endorsing this or that, filling out my spaceship collection, et cetera. I see some interesting space woman up against a window somewhere and I gallop on up to speak to her (note that this is a hypothetical situation, and i'm pretty sure this never actually happened. Let's pretend). She says something with an intricate detail, and thoughts that are real and have value. I can respond with simply a few options that generally involve either probing her further along her established conversational route or otherwise asking her something mission-specific, or saying goodbye. I can't, however, respond with my own tale of boring details and how I was abused as a child or whatever, because this cannot be properly expressed in the shorthand as an option you would want to choose at a moments notice, and would invariably just lead to confusion if it was attempted. 
 
Don't get me wrong, i'm not saying I demand this to be something that future games work on, and I realize that the main character having a long-winded response to every damn space woman would perhaps make the game tedious and probably disliked by the sniveling masses (you know who you are, CODBLOPS fans). I just would like to point out this discrepancy between video games and other forms of story-telling, like novels or movies, in which all characters are interesting and have things of real value to say. In games, it seems all characters are interesting except the one you play.  
 
That's not to say the character you play doesn't have a profound role to play in the world, as I would say it is quite rare that you end up in the shoes of a janitor or cook and much more common to be the person to save the world from definite destruction. From this standpoint, the character is certainly "interesting", but really, it is not because the player decides to make it so. I suppose this is why the majority of games either choose a silent protagonist or one that only speaks in unchangeable cutscenes.  
 
To sum up my point, it is often in games to play the character of someone who is told the long-winded "tales of the forest" or whatever it is by some NPC, but never the player's role to actually tell the tale to some random adventurer.  This seems like an awfully one-sided approach to story-telling, but regardless it seems to be the most interesting, invariably.  
 
I hope at least some portion of that made sense. 

I've been doing a lot of two things as of late, playing Bioware RPG's and thinking. If one is to do this too much, they start to realize some truly odd things about the way the story in these games (namely the Mass Effect series and Dragon Age: Origins) works as far as how much effect the character you play (you referring to the person playing the game) has on the story as a whole, or at least the small details of it. Let me elaborate.  

Let's say i'm playing Mass Effect 2, as good 'ol Commander Shepard. I'm running around on the Citadel, endorsing this or that, filling out my spaceship collection, et cetera. I see some interesting space woman up against a window somewhere and I gallop on up to speak to her (note that this is a hypothetical situation, and i'm pretty sure this never actually happened. Let's pretend). She says something with an intricate detail, and thoughts that are real and have value. I can respond with simply a few options that generally involve either probing her further along her established conversational route or otherwise asking her something mission-specific, or saying goodbye. I can't, however, respond with my own tale of boring details and how I was abused as a child or whatever, because this cannot be properly expressed in the shorthand as an option you would want to choose at a moments notice, and would invariably just lead to confusion if it was attempted. 
 
Don't get me wrong, i'm not saying I demand this to be something that future games work on, and I realize that the main character having a long-winded response to every damn space woman would perhaps make the game tedious and probably disliked by the sniveling masses (you know who you are, CODBLOPS fans). I just would like to point out this discrepancy between video games and other forms of story-telling, like novels or movies, in which all characters are interesting and have things of real value to say. In games, it seems all characters are interesting except the one you play.  
 
That's not to say the character you play doesn't have a profound role to play in the world, as I would say it is quite rare that you end up in the shoes of a janitor or cook and much more common to be the person to save the world from definite destruction. From this standpoint, the character is certainly "interesting", but really, it is not because the player decides to make it so. I suppose this is why the majority of games either choose a silent protagonist or one that only speaks in unchangeable cutscenes.  
 
To sum up my point, it is often in games to play the character of someone who is told the long-winded "tales of the forest" or whatever it is by some NPC, but never the player's role to actually tell the tale to some random adventurer.  This seems like an awfully one-sided approach to story-telling, but regardless it seems to be the most interesting, invariably.  
 
I hope at least some portion of that made sense. 

I think part of how a player makes their role at least fit into the world is through customization, so the more of that the better.  It makes us feel like we're creating at least a piece of the world.
 
But I totally know what you mean, and I'm a bit confused why you DON'T want at least some future games to try out the storytelling option a bit more.  I think the one reason it hasn't been tried too much is because the player-character is sort of a stand-in for you.  And as a player discovering a game, you don't know the facts usually unless you're told.  
 
I think where this might work is when you HAVE accumulated a bunch of different experiences.  You might, say, be in a situation where you were asked what has happened to lead you to the conclusion that the Evil Whatevers are attacking.  You can pick from several story elements to try to reconstruct what happened.  Some games do this as a minigame which is usually inconsequetial, and some find it irritating that they'll either have to take notes or just suffer through it.
 
Another idea might be that you will tell bits of your backstory to impress or enlighten some random person, but it depends on what you choose as to whether or not they'll respond with their own ideas (and if you talk too much, meaning try out all the options, they'll get bored and stop responding, so you have to be smart and pick the option you think they might be interested in).  
 
There was a bit of that, I think, when one of the conversation options for Shepard let you vent about saving the world to one of your teammates, but it was brief.
 
For many, though, too much player-character stuff makes them feel alienated from their own character, especially if it's stuff that they the player are unfamiliar with themselves.
 
Another remedy would just be a more dynamic conversation system than just a few menus from a list, but games have been trying to get that right since near the beginning.  There have been some cool ideas, like picking words out of a list to fill in the blanks in a sentence, and those words you collect over time.  Sort of a combination between a parser and a list of options, with a lot of different possible combinations but a limited amount of times you can try.

"I'm a bit confused why you DON'T want at least some future games to try out the storytelling option a bit more "

" For many, though, too much player-character stuff makes them feel alienated from their own character, especially if it's stuff that they the player are unfamiliar with themselves.  "

If I recall correctly Fallout New Vegas had a few moments where your responses were from things in your characters back ground but nothing major. However in a game like Mass Effect what your saying is more important than in Fallout because in Fallout you are essentially a camera with no past experiences. Having your character tell long stories to NPC would kinda break the immersion though wouldn't it, it would make you fell like your just following some guy around rather than being that guy. 
 
EDIT: The other guy already mentioned it and better than me.