Controlling a character on screen is almost like an extension of yourself. The actions that character can perform are assigned buttons on the controller (or keyboard/mouse for PC gamers) that you control. You expect the character to maneuver like a regular human for a sense of reality, but also perform some hyper actions for a sense of fantasy. A lot of games have mastered the physics behind this and a lot have failed in this regard.
Since a lot of us Generation Y's grew up playing platform games, it's instinctual to have the character jump when you press the action button. It makes sense, there are obstacles you must get around somehow. But even games beyond the platform genre have incorporated jumping as the main action you can perform (besides shoot/attack).
Some games do not require characters to jump. In Zelda: OoT, Link would automatically leap if you run him off a ledge, but you could not directly control his jumping. The GTA/Red Dead Series could make your character hop, but it wasn't integral to gameplay. It was included to give gamers the familiar sense of how a video game character should be controlled.
I would like to see video game physics of a character with more lifelike characteristics. Perhaps you can trip if you run too fast or run over some debris. If you run into a wall, you actually get hurt. Granted there are limitations to our current control system, but game developers are smart. I'm sure they can figure out something.
Also, I think a really cool way to experiment with character mobility would be to create a sandbox game where you can control a variety of animals. Just imagine controlling a snake slithering around, or a cheetah running at full speed, or a large and cumbersome elephant stomping about.
Image source: http://www.emutalk.net/threads/49810-Mupen64-in-C-on-XNA