Which came to be first - bigger airplanes, or longer runways?
It is the larger aircraft that came first. Historically, the runway length (and design) of an airport has depended on the aircraft operating there (among other things), though this is changing. For example, the Prague airport site:


  The main runways – RWY 04/22 and RWY 13/31 – were continually lengthened during post WW II operations which kept them in line with the demands of the aviation industry until the mid 1960s... 


The runway required for the aircraft depends on the take-off weight and the lesser runway length restricts the payload or range. As a result, for economical operation, the runway needs to be longer.

However, there are other constraints to consider:


For large aircraft, the runway (and taxiway) width, apron and parking space are important considerations. In most cases, lengthening the runway will be easier than taking care of these other constraints.
In the end, airports have to weigh the benefits of letting heavier aircraft operate against the resources required for runway lengthening. For major airports (like Heathrow, for example), the cost of runway expansion will run into the billions.
Unless the aircraft is going to be operated regularly (which depends on how many airports it can land), there is no point in lengthening the runway in an airport.
The economic operation of the aircraft has to be considered. For example, the A380 was designed around a 80m 'box', which the airlines had decided is the maximum size they can operate economically.  
A bigger aircraft does not mean more runway required. For example, the A380 requires (slightly) lesser runway compared to 747-400.


If it comes to that, aircraft requiring more runway can takeoff from smaller ones. For example, a Boeing Dreamlifter once took off from a 6000 ft runway (it normally requires > 9000ft).



The airports with longest runways have them for different reasons- the Qamdo Bamda Airport (5500m runway) has a long runway due to its elevation (>4300m above MSL), while the Ramenskoye Airport as a long runway (5400m) due to the aircraft operated there; it was a test facility, for among other things, the Soviet Buran spacecraft.