Unfortunately, your cells can’t fill out census forms, so they can’t tell you themselves. And while it’s easy enough to look through a microscope and count off certain types of cells, this method isn’t practical either. Some types of cells are easy to spot, while others–such as tangled neurons–weave themselves up into obscurity. Even if you could count ten cells each second, it would take you tens of thousands of years to finish counting. Plus, there would be certain logistical problems you’d encounter along the way to counting all the cells in your body–for example, chopping your own body up into tiny patches for microscopic viewing.
So far, there is a document of of a team of scientists who have tried this study, and they have estimated over a huge range from 5 billion to 200 million trillion cells.