Assuming it were possible to fold paper without restriction, the height of a piece of paper would double in thickness every time it was folded.
Since one sheet of paper generally has a thickness of about 0.1 millimeter, folding it 50 times would produce a wad more than 1 x 1011 meters in height, and folding it one more time would make the stack higher than the distance between the Earth and Sun.
This has lead to a challenge to merely fold a paper in half more than seven or eight times. The task was commonly believed to be impossible. Over the years the problem has been discussed by many people, including mathematicians, and has been 'demonstrated' to be impossible on TV.
But then, for extra credit in a math class, high school student Britney Gallivan was given the challenge to fold anything in half 12 times. After extensive experimentation, she folded a sheet of gold foil 12 times, breaking the record. This was using alternate directions of folding.
But the challenge was then redefined to fold a piece of paper. She studied the problem and was the first person to realize the basic cause for the limits.
She then derived the folding limit equation for any given dimension, including both alternate direction folding and for folding in a single direction using a long strip of paper. Using the equations, she successsfully folded a piece of paper 12 times -- four more times than the 'impossible' upper limit!
You can read more about her story in the links below.