What runs a g-shock watch?
The watch runs either on special ASIC (if volume is very high), or on MCU (microcontroller), often the supercheap version of MCU which is mask programmable and which uses very low clockrate (less than 1MHz)

They do have memory, but often very little of it -- a small MCU might only have 16 bytes of memory (as registers), while ASIC would only have enough memory bits to store current time and any applicable alarms.

Here is a link to a very simple watch ASIC: http://www.integral.by/eng/download/1077/IZ6199TSe.pdf . Note that this chip comes as a bare die, 1.4x1.2mm big. It will be encapsulated in the black epoxy blob on the actual watch PCB. 

G-Shock may also run a bigger ASIC or an MCU; it will also be encapsulated in epoxy, but information about it is proprietary and will not be published anywhere.