When should we think about edge caching to lighten the load on our main servers, even past what CDN POPs do?

We usually think of CDNs and their POPs as the main way to get stuff closer to people and take some pressure off our main servers. But I've been wondering about going even further with caching, maybe right at the network edge in our own setup, or even closer. For content that changes a lot but not *super* frequently, or for API answers that can be a few seconds old, when should we cache even more than what a standard CDN gives us? Are there certain apps or traffic patterns where this really helps, more than just making our CDN work better? And what kind of problems might we run into if we start putting caches that close to users or specific client groups?