Is it possible to use multi gigabyte DRAM modules with a microcontroller?
I'm going to say no: the most basic incompatibility is that there's way too many pins on those modules even be able to interface it physically to an Arduino.  Ditto for similar small microcontrollers.  An ARM Cortex M3 is getting closer, but few of those actually have the external bus necessary to interface RAM in such a fashion (I think).

But I'll play along an go further.  Assuming you could interface the chip to the Arduino somehow another basic electrical compatibility problem is the voltages required.  I think these RAM modules use something odd like 2.2V - not present on a vanilla Arduino board or really, any other basic hobbyist microcontroller.

Assuming that was taken care of then the chip would pretty much work like any other external memory to the Arduino.  This is really done all the time - people add external EEPROM to store important constants or SD/MMC cards for storage of webserver logs and such.  Of course, within this context the RAM modules presents no real benefit to the Arduino.  Its main feature is speed and the Arduino (and yes, other hobbyist microcontrolers) typically don't break 25MHz in clock speed.  They're too slow to care about speed.  And storage size isn't a big issue since interfacing to an SD/MMC card is distinctly easier and provides as much space to play with.  

So even if it were possible it wouldn't really be advisable.