I have used IPChicken for several years to quickly figure out what my public facing IP address is. I know that isn't very geeky, but it is very "webby" - maybe I'm more of a webophile than a true geek. But enough of my self-analysis... for me it was all about ease. It is very easy for me to remember "www.ipchicken.com". To tell you the truth I can't even think of another way to look up my external IP address - that's how long I've been using the site.
This morning I was pointed to another site along the same vein, MoanMyIP.com (WARNING: not very workplace friendly, especially if your speakers are turned up). Pretty clever.
I'm sure I'll stick with IPChicken, but it's good to know there are alternatives :)
Note to self: do not visit MoanMyIP any more today - once is more than enough... okay, 8 times is more than enough.