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 (in step i, triangle $A_rH_iO$ where $r$ is the remainder of $i+2$ modulo $6$), 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
