(answering the original question, before the edit)
I suppose it depends on what specific characteristics matter to you. There are plenty of examples of non-government "money." WoW gold, Linden dollars (Second Life money), and Chuck E. Cheese tokens all count as money in some sense.
I don't think any of those examples truly count as "open source money" though. Perhaps the main problem in developing such a money before Bitcoin was how to trust that transactions are legitimate. Typically this would require some sort of trusted 3rd party (so if I believe in Linden dollars gold then I trust Linden to be honest and not to get hacked). Some mechanism needs to be in place to prevent me from altering the balance in my digital account or from making copies of my money files. Trusted 3rd parties can indeed do this, but I don't think that is in the spirit of "open source money." The block chain concept is the only open-source mechanism I know of to prevent double spending without relying on trusted 3rd parties.