Cron script not executing on Mavericks

The environment a cron job runs in is quite a bit different from an interactive shell; it's likely that the script is running, but not successfully. One of the biggest differences are that for cron jobs, the default PATH is just "/usr/bin:/bin", so if you use any commands that aren't in /usr/bin or /bin, they won't be found unless your script either sets its own PATH, or supplies explicit paths to commands. The other big difference is simply that it's not connected to an interactive session, so if it tries to do anything interactive (read from the terminal, etc) that'll fail. Try changing the cron entry to:

*/2 * * * * /Users/[my user]/Dropbox/htdocs/auto-update.sh >>/tmp/auto-update.log 2>&1

...and see if anything informative shows up in the log.

It's tough to tell, but what if you added a second cron job to execute every 5 minutes or so and have it call some system built tool that logs messages to system.log?

0,5,10,15,20 * * * * /usr/bin/logger "cron is working"

That way you'll know that cron is running for the user in question and can focus on either starting cron or fixing your script so it runs in the limited cron environment. (You can look at the wall clock and pick a few times that are coming up soon or even the next few minutes - e.g. editing at 12:34 put in 35,36,37,38 for the minutes to run and save the cron file.)