There are a lot of countries that have no or no longer have a flag carrier, defined by "Government Owned".
However, thought it has been privatized, I would argue BA
the flag carrier of Britain. KLM is still the flag carrier of the Netherlands (actually, the current deal has the current owners of KLM with 51% stake, perhaps until Open Skies takes effect).
In any event, there are only a few countries that don't have some sort of official, flag carrier. The US, some smaller/poorer countries without the money, and a few former soviet-block states. Everyone else has a legacy flag carrier or something official.
Those who fail to learn history are doomed to repeat it in summer school.