The record for most rainfall in a single hour is 12 inches (30.5 cm), set in Holt, Missouri, in 1947. What is the maximum amount of rainfall that could theoretically occur in an hour?
A very moist airmass was in place across Missouri during the afternoon and evening of 22 June 1947, as evident by dewpoint temperatures in the middle 70s across Missouri during the afternoon hours
Advances in meteorological science led Locatelli to speculate that the thunderstorm that produced the historic rainfall of 22 June 1947, did not develop along a classic surface cold front, but to the east of a surface dryline (drytrough) and underneath a cold front aloft that had origins in the Mexican plateau. Regardless of the causative surface and upper air features, a complex of storms at a scale not measured well in the data-spare period of the 1940s developed near Kansas City and moved across Holt, Missouri, depositing 12 inches of rain in less than one hour. Rain continued to fall on Holt after the 12 inch deluge for the next couple hours, but only amounted to an additional one-half inch.
A rare meteorological event occurred that day. I offer a semi-empirical frame of reference that takes into account that the most extreme rainfall rates occur for a short duration (e.g. a few minutes).
In "A methodology to classify extreme rainfall events in the western mediterranean area" by Casas et al., (2004): http://link.springer.com/article/10.1007%2Fs00704-003-0003-x they show rain rates of up to 0.5 cm/min sustained for several minutes (based on a large network of rain monitors in Barcelona). IF that could somehow be sustained for an hour, that would give a rate of 30 cm/hr (which is what Holt saw).
For comparison, the most extreme rainfall rate measured in one minute in the USA is 3.1 cm (1.2 in; see http://wmo.asu.edu/world-greatest-one-minute-rainfall ). Actual rain doesn't continue at such an extreme rate for much longer, but would equate to 186 cm per hour (6 times what Holt, Missouri saw in 1946). The link that kwinkunks provides http://www.bom.gov.au/water/designRainfalls/rainfallEvents/worldRecRainfall.shtml shows a world rate record of 3.8 cm/min which, if sustained, would equate to 228 cm per hour.
The maximum one-hour rainfall may be 305 mm in America, or 401 mm in Australia, but it certainly isn't the world record measurement, which is in the region of 450 to 500 mm (based upon the global rainfall intensity envelope). The maximum possible is governed by three factors, namely 1) the precipitable moisture in the air column, which is a non-linear function of temperature, assuming no limiting evaporation upwind, 2) the dynamics of condensation, assuming no limitation on the availability of raindrop nuclei, and 3) the lateral advective delivery of moisture. This means that the maximum rainfall is likely to be on the windward side of a high mountain in a tropical oceanic island. We will probably never be able to measure the maximum rainfall because, by its nature, it will be in a very windy exposed site, where it is notoriously difficult if not impossible to measure the rainfall accurately. Also, consider the open area of a rain-gauge funnel, and the probability of that just happening to be in the right place to measure the maximum rainfall over many square kilometers. A good description of the probable maximum precipitation is given by Greiser in: http://www.juergen-grieser.de/extremerain_standalone.pdf
According to the World Meteorological Organization, a record 1,825 mm (71.8 in) of rain fell in 24 hours at Foc-Foc (elevation: 2,990 m; 9,810 ft) on the French island territory of Réunion in the Indian Ocean on 7–8 January 1966. The event occurred during the passage of tropical cyclone Denise.