LarryD:
Arduino and LCD/LEDs, not going to happen.
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Well, true if you want to use a watch type battery. LarryD is correct in that an arduino even by itself is much more power hungry than a watch. Depends on how big a battery you are talking and how long you want it to run from that battery. There are a number of processor chips out there (I am more familiar with the PIC ones) that can operate down in the microamp range but you also have to run them with a slower clock (like the 32khz crystal typically used for watches).