What I find interesting about the Saints Row series - which I will now refer to exclusively in the past tense, because by the end of Saints Row IV the story arc has been written into a corner so hard that it left an imprint of its face on both the walls - is that I went along with it completely. For Saints Row 2 I created a character, and for each subsequent Saints Row game I made the effort to recreate the same character as closely as possible, even visibly aging them a bit in each new instalment. Continuing the story as anyone else just didn't feel 'right'.
Let me tell you about him: In my mind, his name was Spider. Because he was very thin, and (obviously) used the Cockney voice that I think sounds equal parts devious and perverse. I'd make his eyes wide and starey, darkening the areas around them with make-up, and slightly turn up the sides of his mouth into a permanent half-smile. Spider decided during the course of Saints Row 2 to model himself on a Batman villain, and from that moment on would always take the earliest opportunity to dress up in a colour co-ordinated suit with matching bowler hat and domino mask. Not that he thought of himself as a villain. He just felt it was time someone reclaimed Riddler-chic.
Saints Row, I think, found the best possible middleground between having a customisable protagonist unique to each player and having a protagonist with meaningful characterisation. The trouble with games like your Dragon Age or your Skyrim is that you choose every action, every line of dialogue (this goes back to what I've been saying about dialogue trees of late), and every moral choice that your character makes, as well as their appearance. The end result being a character that feels like they don't have any personality at all. It's not you, the player, because they're restricted to acting within a fairly small pool of abilities and responses, and it's not a character of their own, either, because it has to double-check with you before it can lift a finger.
The point being, a character we control down to the ground is a character that cannot possibly surprise us. And being able to surprise us is what makes a character interesting. Mass Effect's Commander Shepard is customizable to a large degree and speaks through dialogue wheels, but it's the moments when s/he takes his own initiative that gives him/her that bit of oomph. Like when we give him/her the vague instruction to do something renegade-y and s/he decks a journalist in the face.
Similarly, my boy Spider was always finding ways to surprise me throughout the course of Saints Row. Once you've created their look and style and picked a voice, their decisions and dialogue lines are out of your hands, so the games are full of great character moments. And because we, the player, created the protagonist, there's a greater sense of investment and ownership of those moments, even if we have no control over them. The actions that the protagonist makes are always the same, but a lot of personality can be conveyed solely in appearance and voice. The same action performed by two wildly different characters can have different interpretations.
And it was also pleasant, while I was deliberately growing this character as the series went on, to see the games themselves grow and develop. They were gradually casting off the stigma of the first Saints Row basically just being one of many GTA clones. They moved more towards the outlandish end of the genre, the same area that GTA itself once occupied but gradually shunned, and eventually created its own identity. Which was fitting, because if you'll indulge a little over-interpretation on my part, I think 'identity' could be considered one of the ongoing themes of the series.