Does the Wii Remote controller have any built-in memory ?

Yes, but it is microscopic when compared to most memory solutions. The Wii-mote controller features a paltry 4KB of memory. Seems almost pointless, but consider that that most DS and GBA game cartridges sport even less EPROM space for their save files, many offering only 512 bytes (half of 1k) to record games in progress. Developers still managed within this limitations and they will again be able to work their magic with the Wii-mote's 4k of non-volatile memory.