TamaExpo logout passwords are based on character type. A logout password can be reused as long as you still have that character. The chart at the following link lists the Royal FamiTama logout passwords for 1000 GP prizes.
Click here to see the Royal FamiTama chart. Note: when you click the link, an image will appear scaled to fit your screen. If you move your mouse over the image, the mouse pointer should change into a magnifying glass with a plus (+) in it. (If it does not, click the image first, or press F5.) When it's as such, click the image to see it full size.
Step 1. Determine your main character (the one in the center of your screen, or listed first in the status screens.)
Step 2. Activate the connect icon (upper right corner of your Tamagotchi screen) and select the PC menu entry.
Step 3. When the login password appears, press the B (middle) button. The logout password prompt will appear.
Step 4. Enter password corresponding to the main character, from the chart at the link above. (Note: there are two entries for some characters, because some characters can be either male of female. If one doesn't work, try the other.)
Step 5. Watch as your Royal FamiTama collects a 1000 GP prize.
Note: there are two blank spaces in the table. These are for Jeweltchi (male and female.) They're blank because we don't have a pixel image of Jeweltchi yet.
Thank you, spudilike, for the pixeled characters.