How many times can a piece of paper be folded in half? What is the maximum number of times a piece of paper (of non-zero thickness) can be folded in half (mathematically)? 
Edit: 
I totally forgot to mention the "non-zero thickness" part; I've now included it.
 A: The zero thickness model is a bad model.  For nonzero thickness the problem was studied by Britney Gallivan, who derived an upper bound on the number of folds based on the thickness and dimensions of the paper.  The current world record for actual paper was set by her; it's $12$.  There are some additional details and references at her Wikipedia article.
A: Theoretically you can fold the paper into half infinite number of times since $1 - (\frac{1}{2} + \frac{1}{4} + \frac{1}{8} + \cdots) = 0$. However, practically the number of times you can fold the paper into half will depend on the dimensions of the paper (length, breadth and thickness), the stiffness of the paper, and more importantly your patience and mechanical strength to keep folding it.
A: A very simple constzraint: Folding a paper of thickness $d$ $n$ times produces a block of thickness $2^nd$, hence requires the existence of points of distance $2^nd$ in the original sheet. Typical paper thickness is $0.1\,\text{mm}$, so already by this constraint, folding twelve times requires a paper of length at least $40.96\,\text{cm}$ long (about A3). Of course this would require a "tower" only $0.1\,\text{mm}$ thick, which is impossible - as there are about $2^{n-1}$ layers from the bends vertically! This already hints at the essential $2^{2n}$ factor in the strip length formula.
