I've done a little further investigation,
I navigated to
http://my.site.com/DesktopModules/CuteSoft_Client/CuteChat/SupportService.asmx
using IE on the actual server and invoked SupportLogin
It hasn't timed out using this method (after 20 minutes or so).
Debugging reveals that repeated calls are being made to DNNChatProvider.GetUserInfo for users that are NOT currently on the site. This is a development server and, at the moment, I'm the only one using it.
Currently DNN_Users contains 5478 rows and DNN_SiteLog contains 2,082,999 rows. (This database is a copy of a production database that has been active for about 3 years).
I really, really hope that DNN_SiteLog is not being processed here.
Ah, It just ended with <int xmlns="http://tempuri.org/">0</int>
I did a quick Client Login and still timed out - so it's not a one-time deal.