What are the best services for archiving web resources "on-demand"?

And what about ones that can parse through Ajax code and that can go through a user account log-in?

In particular, I’d appreciate knowing about services that can archive up an entire private forum. I’m willing to pay some money to prevent valuable Internet history from being lost forever.

There’s the Internet Archive, but I’ve found the chat hostile to my intentions.

You’re probably looking for a web crawler. The Wikipedia page lists quite a few, and has a section on dynamic page (AJAX) crawling.