Although turning on CBB is an easy way to deal with the situation, it may actually be making things worse in the long run. You're giving him something to wake up for.
I'm not being holier-than-thou, BTW, I have 100% sympathy for you as I'm in a similar situation. A couple of nights ago 13m ds2 went to sleep at about 7.30, woke once at aroun 11.00, then again at 2am, and after that slept about a total of 2h until I gave up and came downstairs with him at 5.30am. That was a particularly bad night for us, but he wakes 3-8 times a night, and is often up at about 5am. It's a complete killer.
I do try to keep night wakings as boring and as low-key as possible. Generally it's clear that he's looking for something to help him go to sleep (ie my boob!), but sometimes he just goes boing and is ready to play. Then, I confess, I generally give up and let him - like you.
When dd was about the same age, she was a good sleeper and slept through the night, but tended to wake at about 5am. She shares a bedroom with ds1, who could sleep for England, and we didn't want him to get out of his sleeping-until-woken-by-us habit. But OTOH we didn't want to encourage her to get up so early by playing or giving her something interesting to distract her. So we would ignore her for longer and longer. 5mins at first, then 10, then 15, and so on. We would go to her before she started getting loud and annoyed, but never wait for that to happen. Gradually, over a period of weeks, we found that we never had to ignore her for more than 25-30mins, because she gradually woke later and later, until she was waking at 6ish and playing quietly in her cot for 20mins, which was acceptable for us.