For some reason the combination of swfobject.js and script.aculo.us Ajax.Autocompleter on the same page causes the latter to fail. Autocompleter doesn't make its Ajax request. A separate Ajax control on the same page that uses Ajax.Updater doesn't seem to have the same problem.
prototype.js (used by scriptaculous) and swfobject.js might be incompatible. What are the versions of theses tools you are using ? Did you try to switch the order of the 'script' import tags in order to import swfobject first ?
Bah, I should have included versions tried in the original question.
I've tried a combination of swfobject 1.5, 2.0, and 2.1 (current) and both the 1.7.x and 1.8.x versions of scriptaculous, which rely on 1.5.x and 1.6.x of prototype.js, respectively. I've tried loading swfobject both before and after the protoype/scriptaculous libraries, to no avail.
I'm led to believe that there's a fundamental incompatibility lurking somewhere, but haven't been able to find anything about it on the googles, which seems a bit odd in itself.
If you're using Firefox on a local machine, AJAX requests don't work for security reasons.
Either upload to a server, or try something like xampp to easily get a webserver running on your own machine.