-- Should you go back after he broke up with you the "3rd time" ? You should never have gone back after the 1st; and most definately not after the 2nd.
Look, you know what's going on. You sound pretty smart and you have sharp instincts - let there be no doubt that your bf has established a pattern that he won't think twice about repeating for as long as you invite him to.
And yes, you are inviting him to treat you badly by taking him back . Why should anything change since he's able to live how he wants to and you're there in between affairs?
So the short answer is "No", don't go back. Respect yourself more than he does.
The longer answer is ask him to prove his intentions and attend "couples counseling" for at least 4 sessions. Then you'll see the truth without any doubt.
Stay strong and good luck!