The National Library of Australia provide Pandora: an archive of Australian web sites. This is a much more selective collection than the Internet Archive, but there are items in Pandora not in the Internet Archive. In particular, items from Australian government bodies which they would rather people forget about tend to not be in the Internet Archive but are in Pandora (thus justifying the name). The contents of Pandora are included in Trove, which also searches Australian library records and other cultural holdings.
An example of a website an organisation whanted to forget was that for the Sydney Olympic Games. The Australian Human Rights and Equal Opportunity Commission (HREOC) for that the Sydney Organising Committee for the Olympic Games ("SOCOG") had engaged in unlawful conduct by providing a web site which was to a significant extent inaccessible to the blind and $20,000 damages were awarded. After the games, SOCOG deleted the web site. However, there are copies in Pandora, which I was able to use for teaching web design at the Australian National University and to help the Beijing Organising Committee for the 2008 Olympics with the web design.
Today I was updating the notes for my Green Technology Strategies course. After the collapse of the Australaian Government's proposals for a carbon trading scheme the website of the Department of Climate Change was "updated" making many of the documents hard to find. I could not find the External Audit Consultation Paper on the site at all, but I found it in Pandora, using Trove.