Yeah, that's annoying and a problem with text areas throughout the interwebz.
So, I went ahead and banged out a script for this; Merry Christmas!
To install, click here (SE WMD scrollbar killer, install at GitHub).
To preview the source, click here (SE WMD scrollbar killer, preview at GitHub).
This script expands away the vertical scrollbar for all open post inputs. It re-expands, as necessary, when new inputs are opened, or text is typed or pasted.
Go from this:
To this:
Automatically!.
Notes:
- There is no shrink. Inputs are expanded only.
- Note that "WMD" is Stack Exchange's name, not mine.
- Tested on Chrome(like) browsers using Tampermonkey.