"My political opinions lean more and more to Anarchy (philosophically understood, meaning abolition of control not whiskered men with bombs) [...] the most improper job of any man, even saints (who at any rate were at least unwilling to take it on), is bossing other men. Not one in a million is fit for it, and least of all those who seek the opportunity."
~J.R.R. Tolkien
This quote is relatively famous among anarchists. People are always keen to seek out famous or esteemed individuals who share their views.
Despite this quote, I don't believe Tolkien ever brought himself to becoming a true anarchist, but it's clear that his views were tending that way. This is abundantly clear within the first few pages of The Lord of the Rings when Tolkien is describing Hobbit society.
Concerning Hobbits
Hobbits embody everything that is homely and warm about humanity. They have a deep love for family (Sackville-Bagginses notwithstanding), community, gardening, and of course, food and drink.
"At no time had Hobbits of any kind been warlike," writes Tolkien, "and they had never fought among themselves."
At the end of The Hobbit, Tolkien's first published novel in his Legendarium, the dwarf Thorin relays the true greatness of Hobbits in a speech to Bilbo: "If more of us valued food and cheer and song above hoarded gold, it would be a merrier world."
Tolkien certainly wrote Hobbits as something of an ideal for humanity: they love each other, they love nature, they don't go to war, and they don't lead themselves or their friends to near-certain demise on quests for hoarded treasure. It's no coincidence then that Hobbit society is also the closest thing to peaceful anarchy in Tolkien's universe.
Anarchy in the Shire
J.R.R. Tolkien writes at some length about the the community and societal structures in The Shire, the home of the Hobbits. All of the following quotes come from the prologue to The Lord of the Rings in the section: "Of the Ordering of the Shire".
"The Shire at this time had hardly any 'government.'" Allow me to briefly stop to point out how much I love that he, much like 
The Hobbits still held a traditional belief in the civilizing influence of the king, even though "there had been no king for nearly a thousand years". I suppose that much like humans, even Hobbits have a difficult time letting go of the superstition of authority.
"The only real official in the Shire at this date was the Mayor of Michel Delving (or of the shire) [...] As Mayor almost his only duty was to preside at banquets, given on the Shire-holidays, which occured at frequent intervals. But the offices of Postmaster and First Shirriff were attached to the mayoralty, so that he managed both the Messenger Service and the Watch. These were the only Shire-services, and the Messengers were the most numerous, and much the busier of the two."
The Hobbits did have an elected 'representative', but the role of government in Hobbit life was more ceremonial than anything else.
The Shirriffs was the name that the Hobbits gave to their police [...] and they were in practice rather haywards than policemen, more concerned with the strayings of beasts than of people. There were in all the Shire only twelve of them"
I would love to live in a city with only twelve cops... twelve cops left people to go about their business and only bothers with escaped farm animals. That would be nice.
In Conclusion
The Shire isn't a perfect anarchy. There's still a limited government (of sorts) and a police force (of sorts) and a people who believe in the king (even though there isn't a king.) It may not be perfect, but it's a hell of a lot better that what we see in our world.
Reading these descriptions of an ideal world, free from the shackles of a totalitarian regime always brings a smile to my face.