Multiple 240 hour visa free stays?

Hi I'm traveling to Mongolia from uk in April and coming back to uk 3 weeks later from hong kong. In the middle I plan to enter and exit mainland China maybe 3 times.
My question is is there a limit to how many of these Visa free entries you are allowed per Any period of time? ie a month or year , is there a limitation on the amount of time in-between Visa free entries?

Thank you in advance