4.
My daughter did the same thing when she was three. She ate plenty of fiber, and we even tried Metamucil cookies every morning, but she still wouldn't go! The issue became such a problem that her pediatrician was concerned that she might end up with continence problems later, or worse.
So my daughter was prescribed a mild prescription laxative called Miralax. It's an odorless, colorless, tasteless powder that I mixed with her juice once a day. It's non-habit-forming and was developed for extended use. The laxative softened her BM, so it didn't hurt when she went to the bathroom, but also made it all but impossible for her to hold it on purpose. She had no choice but to go.
Once she figured out that she felt better after having a BM, the problem was resolved and she never held it again. We used the Miralax powder for about 6 months.
Be advised. It is a prescription and you must use it as directed, but it's wonderful stuff and now, four years later, my daughter is fine. It might be worth talking to your pediatrician about it.
Best of luck to you.