Which airplane has had the longest production run?
Probably the longest production run of any given craft is the Taylor Cub (more popularly known as the Piper Cub).  Which has, in one form or another, been built since about 1930, moving from manufacturer to manufacturer as the years go by.  There's been a recent resurgence in new manufacturers picking up the design lately because it now can be used in the LSA class of aircraft, created a few years back to encourage manufacturers to produce cheap, easy to fly, short range aircraft for recreational purposes.  CubCrafters is one such manufacturer, and here is their, very current, CarbonCub SS as an example of the type:


Source: https://upload.wikimedia.org/wikipedia/commons/2/25/CC11-160_Carbon_Cub.JPG, Author: FlugKerl2

The longest production run for an aircraft from a single manufacturer is the BeechCraft Bonanza (source).  Which has been in continuous production since 1947:


Source: http://i.stack.imgur.com/utPF8.jpg, Author: Bill Larkins

Another long production run is that of the Antonov An-2 which also started in 1947 and only ceased in 2001, though a variant, a turboprop version called the An-3, is still in production to this day:


Source: http://i.stack.imgur.com/YDvgG.jpg, Author: Sergey Ryabtsev

Other honerable mentions include the C-130 Hercules (in continuous production since 1954, even though only 2300 have been built).  And also, the Cessna 172 and Cessna 182.  Both started in 1956 and are still going strong today, with older airframes still being extremely common.

Also, it is probably worth pointing out the Uh-1 "Huey" Iroquois, which has been in production since 1959, and is probably the longest production run for a rotary wing aircraft:


Source: http://i.stack.imgur.com/jWVEg.jpg, Author: 111 Emergency