How many times can someone refinance their home with the VA Streamline program, sometimes called the VA to VA loan and also known as the IRRRL (interest rate reduction refinance loan)?

As many times as it makes sense.
When interest rates trend downward, as they have been doing for the past few years, it can make sense for someone to refinance their mortgage multiple times within a relatively short period of time.
Example: If interest rates go from 7 percent to 4 percent over a 2 year period of time, it may make sense for someone to refinance their 7% mortgage to 6% and then 5% and then 4%. Of course, it might not make sense, but it might — it really just depends on the numbers. A good loan officer will be able to show you the math equation and show you the savings.
There is also a test that your loan has to pass, called a “net tangible benefit test.” The purpose of passing the test, is to be sure the benefits of refinancing outweigh the costs that are associated with refinancing.

As long as your loan passes the net tangible benefit test each time you consider refinancing, you can feel comfortable knowing the new loan is saving you money each and every time.