The principal institution of the Republic was Alþingi (national parliament, founded 930 A.D.). Alþingi was both a legislative assembly and a court of law, intended for the peaceful settlements of disputes.
But its outstanding characteristic was that there was no executive power whatsoever: no king, no feudal lords, no professional bureaucracy, no police, no standing army. This society was both uniquely individualistic and egalitarian at the same time.
Based on the rule of law, centuries before the Magna Carta, it comes closest to having realized the extreme Neo-Lib. utopia of „limited government“ here on Earth.
This early experiment with governance under the rule of law but without any government authority, lasted for 330 years. Until its demise in the late 13th C. Iceland was a thriving trading nation, outgoing and creative. In terms of politics (the rule of law), literary excellence (the Sagas) and exploration (reaching North-America half a millennium ahead of Columbus) it was highly successful.
Ultimately, it all unravelled due to rapidly increasing inequality – wealth and power being concentrated in the hands of an elite elevated above the law – and due to the absence of effective government to contain it and maintain social cohesion.