While looking for the best application of a Democratic regime, people tend to look either at Athenian Democracy (as applied in Ancient Athens that is) or at Grassroots/Liquid Democracy (as supposedly promoted by (some?) Pirate Parties worldwide). Direct democracy applications are not viable for various reasons, mainly due to the vast number of voters and disinterest by the citizens (that's why most prefer to just vote for their representatives). Grassroots/Liquid is an interesting hybrid between direct and representative democracy. Have there ever been any successful applications of the model, and if not then why?
After the Pirate Party of Germany introduced Liquid Feedback (a variant of Liquid Democracy tools) internally and got a lot of press coverage for it, the Landkreis Friesland is now introducing LQFB as a tool to gather more diverse opinions by its citizens.
As this is still an experiment and the first public one it will be interesting to watch how the project evolves.
Press coverage in German about the initiative can be found at ZEIT Online
The traditional, relatively modern (i.e. post-Middle Ages), example given for grassroots democracy is the historic New England town meeting system (particularly in the colonial period before the formation of the United States of America and when the realities of sea travel limited the British role in governing the American colonies). A closely related system of self-governance, under the Mayflower Compact, is also often mentioned. These systems have largely been replaced with Representative Town Meeting systems.
Protestant church governance in the United States has also historically been radically decentralized, with New England's Congregational Churches, governed democratically at the individual congregation level (i.e. on a church building by church building basis), as an early example, in contrast to the highly centralized organizations of the European established Protestant denominations like the Anglican Church and the Lutheran Church at the time the U.S. denominations or movements were initiated. Many modern U.S. Protestant churches continue to have this form of democratic governance by their members.
Liquid democracy, as I understand it, seems to involve voters providing limited proxies to representatives, as opposed to providing proxies to family members or other trusted people participating in traditional elections in traditional representative democracies on behalf of both themselves and the family members for whom they have proxies (which a number of countries have tried and a few continue to use).
The closest real world example I can think of to the "liquid democracy" concept is proxy voting by shareholders in private corporations, although in practice, it is usually highly undemocratic even as to shareholders. The U.S. State of Oregon considered something along these lines in the year 1912, but ultimately rejected the proposal (which was still crude compared to the issue delineated liquid democracy proxy concept).
It isn't at all obvious to me that "grassroots democracy" as defined at the link in the question, and "liquid democracy" defined at the link in the question, significantly overlap in any way, or can be merged in some trivial and obvious manner. If there is a hybrid system that could be based upon these concepts, the question fails to describe it and it is not common knowledge even among political scientists who work in the area of electoral democracy.