I've had a number of conversations with friends lately about taking over games. In particular, every player in my L5R game has come to me with concerns that their character was "taking over" and getting unfair screen time and plot progression (and damn did that feel good as the GM to hear.) In other conversations it is about one character, and it's usually the player of said character who is concerned about it. The thing about it is, all these characters have one thing in common: they're pro-active.
The GM Is Busy
Running a game is tough. There is no ifs, ands, ors, or buts about it. The job of running a game is a hard one. You have four to six players to entertain. You have a world to maintain. You have rules to keep in mind. You have to craft stories that hook your players, bring the PCs into them, and then position those same PCs into key places to sort things out and you have to do this without appearing to be on rails.
That alone would be a full job, but you also have to progress personal plots, and often when you ask someone what they want from their character they just kind of shrug their shoulders and go "I dunno, loot?"
Why do I say all this? Because you need to know the GM is busy. They're carrying a heavy load. So if you want more time you need to make the load lighter.
The Benefit Of Goals
Even if a player isn't the type to shout others down and claim their time at the table, you know what one thing they can do to make me a lot more likely to get them going in their own plot? Have goals. Having goals is a huge deal in any RPG, but especially table top games where the content is all custom generated. Having the goal of becoming "the grandest duelist to ever live" tips your GM off as to what you want the character to do. It also gives him a lever to hit to ping your character from time to time and see how things are going.
Having a goal gives the GM something to talk about when they're going down the list of PCs to see whateveryone is up to. It lets the GM have something ready to go instead of having to ask what the PC wants to get up to on their own.
Pro-Active Pursuit Is Even Better
With goals helping that much, you can see how you being proactive about your goals can be even easier. Consider the difference between waiting around until the GM decides they have time/space/room to plot out a duelist to challenge you, and you actually just going out and finding a fight.
Now instead of the GM having to come up with a reason and a location for the fight to happen or the plotline to start, they just need to have a "who" in mind. That who doesn't even have to be ready for the first session but can be fleshed out either in session or afterwards when things get going.
This is true for other goals as well. It boils down to the difference between going "well, ummm" when the GM asks what you're doing (or just "hanging at the inn, drinking ale!") and actually having an objective to pursue. In a sense it's almost like writing your own quest in the game, and when the player is running off after their own quest they'll often get the time and attention to see it done. After all, what GM isn't going to want to see where things are going?