How much code can an Arduino board run?
There are several "Arduinos" with memory capacities ranging from slightly less than 32k/2k/16MHz (Flash/RAM/CPUClock) ("Uno") to 512k/96k/84MHz ("Due".)  In addition, there are "mostly compatible" boards that expand that range down to less than 8k/512/8MHz ("Gemma", "Digistump") and up to about 1M/256k/120MHz (TI "EK-TM4C1294XL Launchpad"), and "somewhat compatible" boards with 4G/1G/400MHz (Intel "Edison")

Whether these are sufficient for "full fledged robotics" depends on what you consider that to mean.  Edison is probably the only contender if you wanted to include computer vision, for example, and it probably doesn't do that very well. ("What did you expect at ONLY 1 core at 400MHz?!")

The small Arduinos probably do not support your "very large libraries", either.  For example, while the "Arduino Language" is technically C++, support for the C++ STL is not included due to limitations of the small CPUs used and their run-time environment.

32k of program memory (on the popular Uno) is sufficient for "thousands of lines of code", but I'd think robotics would tend to run toward much more than that.  The small amount of RAM available on most of the flash-based microcontrollers also tends to be a serious limitation when you start looking at standard data-intensive algorithms.  32k/2k disappears quite quickly if you start implementing an extensive user interface, networking, or graphics.  I would say that an Uno-class Arduino is suitable for implementing COMPONENTS of robotics systems (motor drivers, sensor management, etc), but probably not a whole robot.

A good example of large robotics-like software that runs on Arduino-class hardware can probably be found in the 3D Printer space.  "Marlin" firmware runs on an Arduino Mega or equivalent (256k/8k/16MHz) (with RAMPS shield), and there are alternative driver packages that run on smaller and larger arduino-like hardware.  Marlin is about 280k lines of code, according to cloc.

Most Arduino hardware does NOT have expandable memory.  The AVR cpus cannot run code except from their internal Flash, and few of the other boards have any sort of memory bus that could be used for expansion.