There are experiments underway.
Iran is working hard to develop a country that is both theocratic and a representative democracy. Afghanistan and Iraq have also recently adopted constitutions with this objective, and Morocco and Egypt have toyed with the idea.
Nigeria's federal system has given rise to a system in which Northern states in the Sahel have become Islamic governments on a Middle Eastern and North African model, while the Southern states have continued to be Western style, Christian-animist states typically of countries in the relatively early stages of economic development.
Few other countries have maintained this kind of stark divide at a subnational level in a federal system and it isn't clear that the experiment will continue to work, although there are some similarities in Bosnia.
Iraq, and Lebanon's systems of power sharing between competing ethnicities on something other than a raw population basis. Israel and India both have non-uniform family and probate laws particular to the religion of the people involved in these private law matters.
Somalia's new governmental regime is quite innovative with a major focus on indirect election of legislators representing key power factions and on gender balance. South Sudan's new regime has some similarities to this approach.
China has drifted far afield from Stalinist style Communism, in favor of a system with non-partisan elected local officials, a politically responsive one party democracy at higher levels of government, property rights that are recognized but are far less absolute and secure than in Western style legal systems, harsh criminal responses to corruption, and far more control over the private lives of individuals than the West. Some Western conservative politicians have admired the consistent high levels of economic growth that this systems has produced, to its military strength, to its merchantalist trade policies, and to its control of its people. Some Western liberals have responded favorably to its massive infrastructure investments and its alleviation of global poverty at unprecedented rates.
Many countries in China's sphere of influence have sought to, or been compelled as a matter of political reality, to emulate its system.
Cuba started as a Stalinist style regime but has innovated in the direction of very progressive social democratic style policies, but without letting go of its authoritarian aspects.
Countries like Singapore have started more firmly from a Western political economy model but have also borrowed heavily from China's playbook when its comes to authoritarianism and high levels of government owners (especially in the housing sector).
Switzerland's highly federal system with unprecedented levels of direct initiative and referendum style law making remains a unique experiment.
A wide range of systems for selecting government officials, ranging from pure first past the post elections from single members districts, to very pure proportional representation list systems. Some countries have unicameral legislatures, while others have bicameral systems (with or without real power in an upper house). Some bicameral systems have indirectly elected upper houses or appointed or hereditary upper houses, while others have directly elected upper houses. Some countries are constitutional monarchies, some are "republics" (in the sense of having no hereditary officials), and some are absolute monarchies. Malaysia has a rotating council of monarchs. There are European mini-states with co-rules (one democratic and one hereditary), and with a theocratically selected sovereign. Some elections have heavy public funding, others are mostly funded by special interests.
The United States has features like the electoral college and the right to bear arms that are unique to it. There have historically been military forces where rank and file soldiers elect their own officers, and militaries in which soldiers and airmen provide their own gear at their own expense.
Bhutan has done all sorts of atypical things in how it conducts its governmental affairs, such as its focus on a National Happiness Index, rather than it's GDP.
There are only about 200 countries in the world, and many of them have followed the handful of models that were popular when those countries were established, and the institutional and legal inertia make political institutions hard to change, so there are only so many variants that are possibly in existence at any one time.
These are real people's lives at stake too, so decision makers are naturally wary of untried experiments. If someone comes up with a proven compelling innovation somewhere, it will no doubt be copied. But there are lots of risk and there is little reward, in trying something new that might not work, if there is an alternative proven solution that could be used instead.
