A recent question on the Channel Islands made me aware that while they are under the governance of two bailiwicks (Guernsey and Jersey) and are (British) Crown Dependencies, they are not part of the UK or apparently any country.
What other regions exist, if any, that have the property of being governed, but not being part of a country?
This question is difficult to answer because:
In order to not open Pandora's box of international recognition, one could use the Three-Elements definition of statehood, and by extension 'being a country', to define what a country is: a defined territory, inhabited mostly by a defined people and exercising state power over the territory. So you might take a look at the well-defined boundaries of Luxemburg, find a population within that predominantly considers itself Luxemburgish and a central government that decides on and enforces Luxemburgish laws.
But outside of these rather simple cases it often and quickly gets muddy. Take for example Northern Cyprus. There is very little practical debate about the territory, population and state power. However, only one other country in the world officially accepts this de facto situation as the de jure one: other than Turkey, no country officially accepts Northern Cyprus as its own state. A similar case can be made for Taiwan versus the PRC although in absolute numbers more countries have officially recognised Taiwan (none of which can have diplomatic relations with the PRC, however) and many more have sub-official diplomatic ties (like Germany having a 'German Institute Taipei' in lieu of an embassy).
However, just because regions might de facto be independent and not part of a country according to the above criteria, the country in question might still attempt to pass legislation for these areas and attempt to enforce it as much as possible. Lacking direct political (and thus executive) power over the region in question, this enforcement will be either on paper only or insofar as enforcement can be performed in one's own controlled territory. In the linked case, South Ossetian government officials would adamantly refuse to be seen as governed by Georgia but Georgia would maintain it is governing South Ossetia.
Finally, there is the question of where to draw the line. A number of autonomous territories may retain considerable self-determination except when it comes to certain issues (typically defence and international relations) reserved by a national government; irrespective of whether inhabitants of these autonomous territories have the same rights and duties of inhabitants of other parts of the same country. At some point one might have to arbitrarily decide which example still fits the bill and which does not.
That said, there are a number of entries of such a list that would suggest themselves maybe more than others.
The former colonies of the British Empire that have not formally declared independence; as well as the Channel Islands and the Isle of Man.
This includes a number of Caribbean islands, Saint Helena, the Falklands, Pitcairn, the UK bases in Cyprus and Pitcairn; none of these are represented in the UK parliament, all have some degree of home rule and autonomy in internal affairs but all are represented internationally by the British government.
The former colonies of France that are not Départements d'Outre-Mer (Overseas Departements)
This includes the French part of Saint Martin, New Caledonia, French Polynesia and others. They typically have some form of home rule but are represented internationally by France and they also take part in French and EU elections as well as having EU citizenship.
The non-European parts of the Kingdom of the Netherlands
Aruba, Curacao and the Dutch part of Saint Martin (Sint Maarten). Their inhabitants are EU citizens but they do not take part in Dutch national elections. The territories have some form of home rule but the central government retains certain powers.
The Faroe Islands and Greenland
These form part of the Kingdom of Denmark but are not part of the European Union. They do participate in Danish elections although they enjoy a high degree of internal autonomy.
The territories of the United States
Including Guam, the Northern Marianas, the US Virgin Islands, Puerto Rico and American Samoa. Depending on who you ask, the District of Columbia might be included in this list. All of the above do not participate in nationwide elections (e.g. presidential or House) although many send non-voting delegates to the House and/or Senate. They do participate in presidential primaries and their citizens (except Samoans) are full US citizens.
(This list is incomplete. You can help Stack Exchange by expanding it.)
I have somewhat intentionally stopped here as I have listed all overseas territories of western powers that mostly fit the bill. These typically derive from European colonial empires or, in the case of the US, the United States' colonial ambitions. Cases can be made for non-European powers or for European possessions of other European powers as is done in Wikipedia's List of dependent territories; however, I wish to stay away from the potential controversy.
