One example of a simple yet fun project based around an Arduino is the classic game of "Rock Paper Scissors" by the folks from tinkineering. They have constructed a bare Arduino-compatible circuit on some protoboard for their version of the game, which keeps the cost down without forgoing any ease of use. The gane itself is quite playable, and with the sheet over the game's LEDs you can easily keep track of the score and the order of "what beats what" when playing the game, for example:
This is a lot of fun and with a little effort could be expanded or mounted in a more permanent enclosure - either way, check out the project page to get started. And for more, we're on twitter and Google+, so follow us for news and product updates as well.
If you're wanting to make your own Arduino-compatible circuitry without a full-sized board, you'll need an ATmega328P MCUs with Arduino Uno bootloader:
This is the same Atmel AVR ATmega328P microcontroller used in the official Arduino Uno, as well as our Eleven, EtherTen, USBDroid, and other boards. Perfect for building your own Arduino-compatible project directly on a breadboard or on a custom PCB, or for replacing the MCU in an existing board. Comes with the Arduino Uno bootloader pre-installed. Better still, it even has a special label stuck on top with details of the pinout, so you don't even need to look up the datasheet when connecting it up in your project! For more information and to order, click here!