I have a running friend with Achilles problems who bought a stand-up-bike (SUB). Which does not have a saddle or any way of fitting one, here is a picture below.
I am toying with building my own. You buy a cheap bike, and you put on a handlebar extender and take the saddle down as low as possible.
You don't take the saddle off for two reasons: Firstly, leaving a pole sticking up below you is asking for something terrible to happen and secondly, you can't indicate whilst standing up (believe me, I've ridden his bike)