We have the 328Ps in the store. https://www.adafruit.com/products/123
You can program it in an Arduino, then remove the chip to put in your own circuit. https://www.adafruit.com/products/50
The 328P will need some support circuitry. If you are new to microprocessors, it might be better to go with something like the Arduino Micro: http://www.adafruit.com/products/1086
This has all the essential circuitry on-board and you can plug this directly into your computer for programming. We do expect to have a 'headerless' version sometime soon that would be even more compact.
There is also the "Teensy": https://www.adafruit.com/products/199
which is a very compact solution as well. Not quite as beginner friendly as the Micro, but easier than working with bare chips.
Both these boards are designed to run on 5v. That would require 2 cells and a voltage regulator. I suspect they might be able to run on a single 3.2v cell (the Arduino UNO does), but I have not tested this.
We have both 3mm and 5mm white leds.https://www.adafruit.com/products/778https://www.adafruit.com/products/754
You will also need 9 transistors for switching the rows of LEDs. These should do the job: https://www.adafruit.com/products/756
We do have these https://www.adafruit.com/products/970
which pack 8 transistors into a small package. 2 of these may be more compact than 9 discrete transistors.
You will need some 1K resistors between the processor and the transistors (sorry, we do not carry these).
The smallest LiPo cell we have is this one: https://www.adafruit.com/products/258
There are smaller ones out there.