How many times can a tourist enter China with a Chinse visa? General speaking，there are three types：China issues single, double, and multiple-entry visas. A single-entry visa only allows you to enter China once, a double-entry twice, and a multiple-entry visa enables you to exit from mainland China and re-enter on the same visa for multiple times during its validity.
Once you leave mainland China, you must either have another valid entry on your original visa or obtain a new visa to return. Remember that even a few hours in Hong Kong or Macau is an exit from mainland China in this context. Also, remember that you cannot gain extra entries on a visa once it has been issued. In case you need more entries, you would have to apply for a new visa.
When filling out China Visa Application Form, you can tick the intended number of entries and validity of your visa in Section 2.2. For US, Canada, and Argentina citizens, they can also tick the last option “Other (Please specify)” and express their wish for getting a 10-year China visa with multiple entries in the blank after. However, you are not guaranteed to get the intended number of entries you select. The consular officers may judge by your actual itinerary and make their own decision. It seems that they are less willing to issue multiple entries if you apply outside your home country.