I would be tempted to use a 555 to drive a 7490 (or other counter) then to look at the counter outputs as well as the 555 clock. If it's a decade counter you would see a binary 0000 to 1010 then back to 0000 on it's outputs; this would be something more interesting to look at. You could set up various triggers to get a feel for how a trigger affects the samples; look at different sample depths etc.
The more you understand the logic analyzer (including it's limitations) the better you will be able to use it.
Mostly though make sure you're having fun with it as you learn how to use it. :)