The world's highest recorded air temperature is officially recognized by the World Meteorological Organization as 134°F (57.6°C) recorded at Death Valley, California, USA on 10 July 1913.
Note that this is in recorded history. Higher temperatures have occurred, of course, at different times during the 4.55 billion years of Earth's history.
El Azizia, Libya, held this record for 90 years, after recording a temperature of 136°F (58°C) on 13 September 1922. It was coincidentally also on 13 September of 2012 that this record was stripped by the World Meteorological Organization after a team of experts determined that there were enough questions surrounding this measurement that this temperature probably did not occur.
The temperature had been suspect in atmospheric science circles for a number of reasons. One being that the time of year is inconsistent with such a high reading. Also, the type and exposure of the measuring instruments cast doubt on the accuracy of the data. However, other temperatures in the same general area approach that maximum, especially in the cloudless southern Sahara, far from the moderating effects of water. Several links are provided below for more information on this process.
Other Earth Temperature Highs:
The modern, most reliably recorded air temperature in the world was 129.2°F (54.0°C) at Death Valley on 30 June 2013.
The highest naturally occurring temperature (at Earth's core) is higher than the melting point of iron and is estimated to be approximately 5000°C.
The highest temperature ever created in a laboratory experiment: Scientists, using the Z machine, have produced plasma at temperatures of more than 2 billion degrees Kelvin (3.6 billion degrees F) at Sandia National Laboratories, located near Albuquerque New Mexico.
Dasht-e Lut, a desert in southeastern Iran, was identified as having the hottest surface temperature (not air temperature) of 70.7 degrees C (159 degrees F) This was only during the years of study in 2004 and 2005 by MODIS, which is a satellite remote sensor, mounted on NASA satellites Aqua and Terra.
Caveats to the Above:
Modern measuring methods, instruments, and techniques are more sophisticated and standardized today. Example: The World Meteorological Organization, recommends that air temperatures be measured at a height of 1.25 to 2 meters (which is approximately 4 feet, 1.2 inches to 6 feet, 6.7 inches) above ground level.
The most likely places on Earth for record high temperatures are in depressions in desert regions, especially in areas below sea level. The Dallol (Danakil) Depression in Africa (Ethiopia), Death Valley in USA, and the area around Lake Eyre in Australia are likely candidates. However, the Gobi Desert's temperatures, while far from any ocean, are mitigated by altitude. The Dallol Depression had a weather station for a short while (only a few years). It was run by a mining company, and wasn't there long enough to measure an extreme maximum to beat the Libyan record. It did however, measure very high mean average temperatures while it operated.
The thing to remember about very hot places is that data is sparse. This is because very few people with high levels of technology stay in these places for long. The environment of the Dallol Depression is hostile to human life. 135