Nice facts.
It's a cool one isn't it?. Shows the power of powers (no pun intended).
The thickness doubles every fold. As Ekim says typical paper thickness is 0.1 mm, so the thickness of the folded piece is
0.1 mm * 2^n where n is the number of folds:
0 folds = 0.1 mm
1 fold = 0.2 mm
2 folds = 0.4 mm
3 folds = 0.8mm
5 folds = 3.2 mm
10 folds = 102 mm = 10 cm
15 folds = 3.3 m
20 folds = 105 m
25 folds = 3.4 km
30 folds = 107 km
42 folds = 440,000 km, more than the distance to the moon.
Of course it would be a very, very thin column by then

edit: it's almost the same equation but with a minus sign. If we started with an A4 piece of paper (297 mm) and folded it lengthways every time, its length is
297 mm * 2^-n where n is the number of folds. After 32 folds that's 0.07 nm, similar to the atomic radius of hydrogen. It would be a column of single atoms. I.e. you'd need a bigger sheet of paper to reach the moon than A4.