When does statelessness feel more like a nice idea than something you'd actually build?

Everyone talks about 'design for statelessness' like it's the holy grail for scalable systems. The concept is straightforward, you know, no session affinity means easier scaling and better resilience. But honestly, I've noticed subtle stateful parts sneak in, or sometimes you just can't avoid explicit state without totally overdoing the engineering. Think about long background tasks, WebSocket connections, or even tricky caching setups, they all can bring in a bit of state. What have you actually seen out there? When has sticking to statelessness become a real pain, and when is it truly worth all the extra fuss it creates in other areas?