The human body consists of between 10^13 and 10^14 cells. (The number another answerer gave of 60 trillion would be 6 x 10^13.) It is difficult to measure the exact number, and of course it would vary somewhat from person to person.
The next question is, how many molecules are in each cell? The size of cells varies, from tiny granule cells in the cerebellum to a large fertilized ovum, and the size of molecules within the cell varies from tiny water molecules consisting of just two hydrogen atoms and an oxygen atom, to enormous DNA molecules consisting of hundreds of millions of base pairs.
A cell is about 70% water, and because the water molecules are so small, that 70% water represents 99% of the molecules in a cell. The next largest contingent of molecules are the protein molecules, making up over 50% of the dry weight of the cell or 15% of the wet weight, and those proteins have a molecular weight of between 1000 and 100,000 daltons.
There are about 10^14 atoms in a typical human cell, which is coincidentally about the same number as cells in a human body.
So, if 70% of those atoms are in water molecules, with three atoms per H20 molecule, that's 2.3 x 10^13 water molecules.
Because the size of the protein molecules is so much greater than that of the water molecules, the amount they contribute to the number of molecules in the cell is miniscule. For instance, the typical liver cell contains only about 8 x 10^9, or 8 billion, protein molecules.
That's less than one tenth of one percent of the number of water molecules, above.
There are various other molecules in the cell besides proteins and DNA -- polysaccarides, lipids, and RNA, for example. Some very large molecules -- macromolecules like DNA and some very large lipids -- are only present in small numbers. But various small molecules are present in large numbers.
But the abundance of these small molecules still doesn't begin to approach the abundance of water molecules.
"The small molecules ... are abundant and diverse. ... Because they are so much smaller than the macromolecules, a very small mass of them consists of a very large number of molecules. If we assume an average molecular weight of 100, then an average E. coli cell would contain approximately 70 million small molecules."
Now, an E. coli cell is a lot smaller than a human cell, but even if that 70 million were 70 billion (as it would be in a cell 1000 times the volume, or 10 times the size in each of three dimensions), it is still dwarfed by the 23 trillion (2.3 x 10^13) water molecules, above.
So if we take roughly 2.3 x 10^13 (23 trillion) as the number of molecules in a cell, and roughly 5 x 10^13 (50 trillion) as the number of cells in the human body, we get approximately 10 x 10^26, or 10^27, or one thousand trillion trillion molecules in the human body.