Heads up, the mods may well flag this as a violation of the subreddit rules - if so, you should consider asking this again in tomorrow's Friday Free-For-All thread.
San Francisco between 1847and 1850 was pretty bad.
Sam Brannan's San Francisco empire during the gold rush made him the defacto law in the city until California was made a state in 1850 and the federal government got involved. He was essentially a mafia don, with control of all supplies coming into the city, and control of the bars and brothels that had spring up to service the huge number of people coming west to seek their fortunes.
As /u/prufrock451 predicted, this has been removed. While it isn't exactly a throughout-history question (it's specific-ish in time, but not geography), it is a poll-type question. As /u/Prufrock451 suggested, you may want to ask this in the Friday Free-For-All, or suggest it as a Tuesday Trivia topic.
I think this is going to be quite a subjective ranking, but Wild West towns during mineral rushes are certainly strong contenders. I know mainly about the mineral rushes in Colorado and then the Yukon Gold Rush, but I'm sure that a look into the history of the 49ers in California would reveal similar places that briefly were truly lawless. These were places that sprang up nearly overnight -- the hasty construction of the town of Creede, Colorado, was so poor that the entire town burned one night in 1892 -- that had no customs, family structures, religious or other charitable institutions, etc. already established in them. They attracted desperate people hoping to make a fortune, often in a get-rich-quick way, and then the inevitable businesses that followed: brothels, saloons, and gambling parlors all appeared to part the miners with what little money they might have made. With new people pouring in every day -- the trains coming into Denver's Union Station in the 1880s were packed, day and night -- there were always new arrivals ready to be fleeced. Official institutions like sheriffs couldn't keep up with the influx of new people to hitherto deserted or sparsely-inhabited places, and politicians were often easily corrupted with the promise of getting rich quick.
The infamous Soapy Smith -- whose wikipedia tagline really sums him up, so apologies for the wiki quote:
Jefferson Randolph "Soapy" Smith II (November 2, 1860 – July 8, 1898) was a famous con artist, saloon and gambling house proprietor, gangster and crime boss
-- personifies the kinds of activities that were common in these frontier towns. He worked out of Denver, Colorado, Creede, Colorado, and lastly Skagway, Alaska, during mineral rushes in those towns. He notoriously controlled Denver politics from the late 1870s until he moved to Creede in 1892. He gained his nickname "Soapy" for a scam he ran involving the selling of soap on the streets of Denver. Later, he operated out of the Tivoli Club at the corner of 17th and Market (not to be confused with the current location of the Tivoli Brewing Company on the Auraria campus). You can see some cool old photos of it here. The Tivoli Club was a gambling den where the house always won, and because Soapy and his associates controlled Denver city elections, it was allowed to operate, despite strong protest from the press, who dubbed it the "slaughter pen" because of how much unchecked violence took place there. In fact, one thing that had originally drawn Soapy to Denver in 1879 was its lax restrictions on gambling.
Eventually, however, Soapy's operation became too well known, and the politicians he had previously controlled were forced by public/media outcry to pass stronger gambling laws. So, in 1892, Soapy headed south to the town of Creede, Colorado, which was in the midst of a silver boom. The town went from a population of 600 in 1889 to more than 10,000 people in December 1891, and was famous as a place where it was never night -- it's in a narrow canyon, and the lights from the saloons, brothels, and other gambling establishments going all night reflected off the canyon walls. Soapy quickly established himself as the crime boss in the town and used that power to try to run out other outlaws. The sheriff, William Sidney "Cap" Light, was Soapy's brother-in-law.
As I said above, Creede burned down -- going from this to this (sorry, can't embed that last image). There's a great image in the Denver Museum of Nature and Science Rocks and Minerals section if you ever get a chance to go there that shows that main street in Creede jam-packed with people. After the town was destroyed, it was quickly rebuilt, but a silver panic in 1893 combined with Denver loosening its gambling laws yet again put an end to this lawless, frontier-period there as train-loads of eager miners stopped arriving and the criminal element, including Soapy, migrated back to Denver.
So there are two examples of essentially "lawless" places for certain times and places in history. I can tell you more anecdotes about similar towns, like Skagway, Alaska, where Soapy died in a gunfight four days after appearing in a parade with the mayor, or Leadville, Colorado, briefly the most populated place in Colorado during the silver boom and where Doc Holliday was exonerated by a jury after shooting the local deputy dead over $5. But beyond just anecdotes, I think what unifies these stories is the conditions, like I described above, that allowed lawlessness to flourish -- and note that they are not stable situations and that once the mineral rushes had ended, they became lawful towns once again.
If you'd like to read an amazing piece of historical fiction set partially in Colorado during this time and a little later, I highly recommend Thomas Pynchon's Against the Day.
All that being said, I'm sure other lawless locations include places in the aftermath of wars, or even during them (like in Syria right now) -- any time that normal society is interrupted, often by an influx of large numbers of desperate people and people who are ready and able to take advantage of them (two groups which presumably have a lot of overlap).
