Thanks to Crossroads for putting me up to wire-wrapping as an alternative way of putting projects together!
Here is my wire-wrapped (minimal) board:

There is a 16 MHz resonator, a couple of decoupling capacitors, a 10K pullup for the reset line, and a 0.1uF capacitor for the reset activation. Also an LED and limiting resistor. Programming is via the 6 FTDI pins, and the power pins are next to that.
Reverse side:

The red wires are +5V, the black ones Ground, and the blue ones data.
Not a drop of solder was used in assembling this.
Tools used were:

- WSU-30 wire-wrapping tool, as recommended by Crossroads
- Wire-wrapping wire in three colours
- Single-inline wire-wrap sockets for the capacitors, etc.
- Dual-inline sockets (2 x 14 pin) similar to the one in the photo
- Extra-long pin header strip for the FTDI connection
More information about wire-wrapping:
http://en.wikipedia.org/wiki/Wire_wrapI know it's a bit of a mess, but this was my first attempt at wire-wrapping. It works, though. The "blink" sketch uploaded and worked (I had to change it to use A5 because I got the pins a bit mixed up when I was wiring up the LED).
This demonstrates that you can get an Atmega328 working, and programmable from the Arduino IDE, with a minimal amount of components, and not even needing power (for soldering anyway).
There is no voltage regulator, that could be added without too much extra trouble.