When you joined
It seems that Duolingo doesn't display when a user joined the site. So most people just guess, or check their browser's history. A few just accept the fact that they can't figure it out.
At first I just ignored that problem. But when somebody posted a discussion asking how to find their join date, I decided to check and find a solution.
Some of you already figured this out, but for those who don't know yet, here's the solution: check the source code. Oddly enough, you can find your join date in there! What's more, it doesn't matter which Duolingo page you start on - any page will work.
To find it, simply right-click and select "View page source". Then hit Ctrl+F and search for "creation_date". The join date will be right there.
Anybody interested in comparing join dates? It would be interesting to see who has the earliest join date. (I have 2013-08-07, or August 7.)
EDIT: Earliest user found in the comments: Mirima, 2012-03-21
This no longer seems to be available (2018-07-29)
EDIT You can still check this via Settings (hover your profile pic in the top-right) → Profile (sidebar) → There's an entry on the left called "Learning since"
I have been practising Japanese with Duo since 2017-06-01; 1 year, 1 month and 28 days now! (though I haven't for several months)
Thanks Timothy. My start date: 2018-01-11T21:04:31. Awesome. I was really curious.