One of the most recent mythical places, the Lomar Republic is a virtual country, created on the Internet in 1995. It is one of several "micronations" that exist. Others include: the Principality of Corvinia, the Kingdom of Talossa, and the Dominion of Melchizedek.
The Lomar Republic issues passports and claims to have about thousands of "citizens." The location of Lomar is sometimes given as between Alaska and the North Pole, although it actually has no geographical location. You can search for websites about the Republic of Lomar- their own site has been down lately.
Unfortunately, some people have tried to take advantage of this novel idea. There have been several scams in Nigeria, where hopeful citizens who want to leave the country have been sold passports applications.