What is the biggest airplane that does not need a paved runway?
Military transports are designed to be operated from unpaved runways- by extension, any of these converted to civilian use can operate from unpaved runways. 

For example, the Antonov Airlines operates a number of Antonov An-124 'Ruslan', which can and do operate from unpaved runways regularly. The Volga-Dnepr group specifically states:


  Multi-leg landing gears equipped with 24 wheels allow to operate the aircraft on unpaved runways ...