This is the task of 404PageFound.com. Imagine a Wayback Machine that isn't a museum so much as it is a directory. WM archives instances of old defunct websites, but 404PageFound finds old websites that are still in operation and serving their original purposes.
"Even large sites (CNN, Yahoo!, and Business Week, to name a few) have old pages that remain in their original layout and have escaped deletion from the server," explains the 404PageFound project page. "For reference, there were an estimated 100,000 websites in 1996, and close to 1 million in 1997. Many of these still remain in their original coding, although the vast majority have been removed or completely renovated. 404PageFound strives to excavate and display these remaining buried gems."
As it turns out, there are a whole lot of these gems. Examples include TinyTIM (tim.org, 1989), aka "the world's oldest running MUSH (Multi-User Shared Hallucination)," the preserved Clinton-era US State Department website (state.gov/index.html, 2001), the static white void of something.com (1999), and the website of finance multinational Berkshire Hathaway, which, while updated as recently as 2013, is based on 15 year-old HTML standards. Some of the sites are making a point, some are just genuinely outdated.
Nebcorp.com, for example, is on a mission to be the longest-running non-redesigned site, while others are legit anachronisms, like Cold Fusion Times. The latter also happens to be hosted at the unfortunately named STD.com (aka the World), the "first public dial-up ISP" on the net.
Anyhow, you could and probably will spend a whole lot of time here. Enjoy.