How many cells does the human body have?

Many. It depends on the person dimensions.
Intro
The only way I can see to tackle this problem is being a physicist. First of all this is a quite difficult question it could take a whole scientific paper to make the right estimation likely it already exist out there.
Method 1: weighting to know
The first way is comparing the weight of a person with the weight of a single cell:
##N=P/C##
Where: N- number of cell; P- person’s weight; C- cell’s typical weight.
Method 2: measuring to know
The second way is by calculating:
##N=P_V/C_V##
Where: N- number of cells; PV- person’s volume; CV- cell’s volume.
Final remarks
I leave the math to you. A third way is comparing the surfaces instead of volumes in method 2 thus you need to make the calculation for each organ. All these methods will give you an approximation quite bad but it is a good starting point.
Some calculationsCell weight[1]. ##3.5*10^-9## Assumptions . All cells are the same e.g. bones brain hair…
The majority of the body mass say 90% is compacted within cells.
Take a person of 70 Kg therefore ##2*10^14## cells. Just to compare the world’s population is estimated as ##7*10^9##
Limitations. this calculation is for healthy people some medical condition such as obesity may alter the cell natural properties such as weight. Further we took as granted that all cells are the same wrong! but useful.
There is a high probability that this number is smaller than the real value just in the brain we have ##1*10^10## [2].
Cited references
1. Molecular Cell Biology. 4th edition. See http://www.ncbi.nlm.nih.gov/books/NBK21473/. Accessed on 09 02 2016.
2. List of animals by number of neurons. https://en.wikipedia.org/wiki/List_of_animals_by_number_of_neurons. Accessed on 09 02 2016.