That's an interesting question... How many angles are there? I would guess a near infinite number, or at least on a scale that is far past countable. I can see nearly seven thousand from where I'm sitting, and the universe is pretty big. I'm going to make the assumption that the mean of all angles is around 45 degrees, though I can't justify that at all. Altogether then, the sum of all angles must be 45 times infinity, which I believe is... infinity.
Unless you made them all coterminal, and just let it loop in circles from zero to 360. Then your answer is fourteen.
However, it sounds like you're asking about this.