I loved watching those numbers go up between visits! As a kid, I would naively think: "This many new people experienced this as well". Of course thinking back, many of those could have been repeat visitors, but it's still so cool to think about.
I remember being so ridiculously proud of myself when I realised I could make a cgi script to write each visitors IP to a text file then on each new visit collate that text file to display a counter of “unique visitors”, then display it next to a normal “hit counter”.. I thought I’d invented the idea of tracking unique-vs-visits. Then I got a job as a web dev for a big(ish) local company and realised every major website was already doing this and just not displaying it.
I've kept those alive at w3counter.com since 2004 (before Google acquired Urchin and turned it into Google Analytics). It's not exactly an in-demand feature so I don't think it's even mentioned on the homepage, but after signing up, you can choose from a couple dozen counter styles, or the standard invisible tracking tag. Been working on redoing that site to re-launch it as a cookieless, zero-personal-data tracker later this year.
Actually I still host https://www.websitegoodies.com/ which I originally made in 1999, and that has a simple hit counter service too. So there are some counters I've been hosting for over 20 years.