Lets start will a typical A4 size paper roughly 300 mm x 200 mm (with a thickness of 0.1 mm) and keep on folding the longer dimension in half. Now you realize that there are two factors that make folding a piece of paper hard :
1. the length and breadth of the paper : smaller dimensions are harder to fold
2. the thickness of the paper : thicker sheets are of course harder to fold
Now, if we fold a paper 7 times alternating the direction of fold everytime,
the length and breadth of the surface will be reduced by factors of 2^4 and 2^3 respectively while the the thickness will increase by a factor of 2^7
So after folding an A4 sized paper seven times,
length of the surface = 300/2^4 = 1.8 cm
breadth of the surface = 200/2^3 = 2.5 cm
thickness of the structure = 0.1*2^7 = 1.3 cm
Now as you can see, the height of your resulting paper is rather significant in comparison to the remaining length and width at your disposal. You can start with a larger paper but the main problem here is the thickness and it will need immense strength to add one more fold manually.
Britney Gallivan has done a lot of research on this area and concluded that in order to fold a sheet of paper in half, it must be PI (3.1415926 ...) times longer than its thickness.
If you had a big enough piece of typically thick paper and folded it over 103 times, it would be thicker than the obervable universe, 69 billion light years. That's exponential growth for you.