What countries in the world ARE NOT formally recognised as legitimate countries according to the UN?

Answer 1

1. Fairly recently, the BBC produced a short documentary series called "Holidays In Places That Don't Exist". Five countries were visited (the year of their declaration of independence is shown in brackets):
(a) Somaliland (1991)
(b) Trans-Dniester (1990)
(c) Taiwan (1971 - derecognition by the United Nations)
(d) South Ossetia (1992)
(e) Nagorno Karabakh (1991)
http://news.bbc.co.uk/2/hi/programmes/this_world/4491059.stm
Note that these are fully functioning states, with governments, armies, border/immigration control, income tax and the whole shebang. They are not just jokey micronations like Sealand, and they are not officially recognised internationally.

2. To that list could be added:
Turkish Republic of Northern Cyprus (1974) - only recognised by Turkey.

3. Here is the current list of United Nations member states:
http://www.un.org/Overview/unmember.html

Answer 2

Micronations. Kurdistan. Republic of China.