Would it be possible to patent Bitcoin?
To claim a patent you must be able to prove against contest that you invented the device, and have a working example that if built to the specs provided, runs. 

As it was this Satoshi who invented bitcoin, and there are prior examples of a digital currency, it would be very hard for any one person to claim a patent so far after the fact. Some one could try, but the patent industry is built to make it hard to write a patent, easy to contest. Admitedly many companies do just that and are good at it, but anyone patenting bitcoin would have a hell of a time proving first invent.