Anthropologist Robin Dunbar said that a person has a limit of about 150 social relationships. More specifically, it could range from 100 to 200 interactions depending on how social the person is. Our network of acquaintances may be three times as large as that of true friends.
Fun fact - there is a neurological condition called hyperthymesia in which someone will remember the name of every person he/she has ever met. In this case the number would be infinite! But for the rest of us, the number of people we can remember will be around 150.