Mario's Classic Moves
As demonstrated by the below videos, Mario's athleticism is truly remarkable, but if someone as athletic as Mario, perhaps a professional or olympic gymnast attempted his moves in the real world, would they be able to perform them, even as a life-time goal?
As for Mario's classic jumps, the answer is: not without a trampoline. In this first video above, Mario does a jump which gives him a nice height above the ground, but after landing, he is able to jump even higher, and upon landing once more, he is able to jump even higher, so high in fact, that he is able to do incredible flips before touching the ground again. Since he is jumping on solid ground and it doesn't appear as though he is flexing his legs any differently, there is no physical explanation for why Mario is able to jump higher during his second and third jumps than on his first jump unless the ground surface conserves energy from his landing and releases the force back up, allowing him to jump higher, like a trampoline.
While Mario is in the air, a quick joystick movement in a particular direction will influence Mario's position in the air. This would easily be physically impossible because in order for an object to accelerate, there must be a non-zero net force acting on it, and in this case there is not.
The set of moves demonstrated in the above video are his fantastic drift-slide-jumps, where with a running start, he is able to belly-slide on the ground, and then push off with his hands (which are in front of him) and flip to his feet with a velocity faster than his original running speed. The reason I would consider this physically impossible is because of the amount of strength required to perform such a move. A gymnast would have to push off the ground at just the right angle to achieve substantial vertical and horizontal velocity, which in this scenario is impossible.
The third move is the side-jump, which results from when Mario tries to jump in the opposite direction he is facing. This move appears almost like a completely-in-air cartwheel, which would require remarkable leg strength to perform without a running start. Also, Mario appears to be jumping higher than perhaps 4 of his own body-lengths, or higher than 23 feet, which is impossible!