Tom,
I just registered on your site without issue. Then, I logged out and back in again without any problems. Is there any specific way to recreate the problem? Is it only on registration, or is it on login as well? As for recommendations, I'd try:
* Disabling any caching you have enabled
* Turn off any JFBConnect profile plugins
* Turn off SEF URLs
None of those should be required permanently, but if you disable one of the above and things start working, that will really help us narrow down a possible cause and come up with more specific solutions.
Let us know if we're testing properly or not, and we'll go from there.
Thanks,
Alex