Yes, I remember a cute cat living on my desktop, chasing my cursor. Somebody on mastodon also linked to https://kickassapp.com/ - an asteroid game where you destroy DOM elements on websites, a project from 2011.

Somehow the web got very serious lately..

Most probably it was Neko toy.

https://eliotakira.com/neko/

https://github.com/eliot-akira/neko#readme

I've definitely been enjoying the sort of mini-comeback desktop toys have had in recent years as little JavaScript scripts on tech people's personal blogs (like oneko.js: https://github.com/adryd325/oneko.js/). If I was a little more cold-hearted I might find it annoying and distracting, but I just love when folks do silly stuff on personal websites to make them feel more "personal" like they did in the early-mid 2000s.

This is implemented with just the right amount of delay/speed to be cute and not annoying. If it was flying around faster and closer to my mouse cursor, I would hate it.

More directly, I recall there used to be a website that allowed you to unit another website and then destroy/deface it with multiple tools. Lasers, bombs, things like that. Disappointingly I can't seem to find a trace of it.

Further nice early examples are xroach and xeyes.