What's the shortest? No idea. Doesn't seem answerable.
You get better at what you practice. Compared to blitz, 15|10 feels like an eternity. But in a difficult position if I want to calculate two different candidate moves, and compare which is better (basically this is analysis) then 15|10 is very fast. I'd say you don't really have time to do this even once for any difficult position.
I have a lot of free time right now. Dan Heisman says that I will improve faster by playing slow games, not blitz.
Chess.com has 30/30, 45/45, and custom. Right now I like to play 15 10 for my "slow games". Those extra 10 seconds per move makes a big difference in how I think, as opposed to blitz.