There is an example for a triangulation with just one obtuse triangle in each step.![enter image description here][1]


In the image, only triangle $A_1A_2O$ is obtuse. With your algorithm we draw altitudes and reach $H_1,H_2,H_3,...$ and always we have only one obtuse triangle, so we are forced to draw it's altitude.

![enter image description here][2]


Red segments are the altitudes and green segments are the added segments to regain a proper triangulation. So there is a triangulation that never terminates.
  [1]: https://i.sstatic.net/lt92b.png
  [2]: https://i.sstatic.net/yDuQO.png