How many times Doctor regenerated?

I posted before that in show he said he's 903 years old. As far as I know he's 10th Doctor. But the number does not make sense to me...

I just watched my old tape I recorded of 1996 movie Doctor Who. In that movie, 8th Doctor said he has 13 lives. Something like that. If the Doctor regenerated for 13th time... what happened to 5 "missing" Doctors? Or does that mean he have 5 lives left?

But I came cross that the council is gone.. Gallifrey was burned. So that council no longer control over his lives. That part he could have many lives as he wants.

Well, it's up to BBC whether they want to continue to the end of 13th Doctor or beyond.

The reason was this question is of David who played as 10th Doctor since 2005. It would be nice to see few more years of 10th Doctor. It would be sad to see him go if they are planning to get 11th Doctor. I don't know. I can't imagine. If it is true that one day he leaves the show, I don't know if I could watch. Like I used to watch 4th Doctor, I didn't watch any after Tom. That means I would have to bid farewell. This sucks!!!