I have only been on HRT one year and definitely not taller now.  Actually my height dropped far too much prior to HRT.  
In my 20s I was 5'11" (180cm).  I hadn't bothered to measure for many years, always filled in forms that way.  I started seriously considering transition at age 53, measured my height and realized I was 5'7" (170cm).  That was quite a shock... I had not been to a doctor in more than a decade.  I immediately started looking for a GP and an endocrinologist, ordered blood tests and began the medical process of transition.  My endocrinologist ordered a bone density test to set a baseline, I'll re-measure that in the next year or two.  My only chronic health issue had been occasional back pain and that has now stopped.  
I wasn't surprised to confirm my natural hormone levels were off.  I had a choice: take testosterone or estradiol.  Easy decision.