Are there any iOS apps that let third-party developers extend functionality? If so, can you give examples? What I'm looking for is any apps that let third-parties write code or modules/"mini apps" for extended functionality that a user can install/execute from within the app.
Nope. As a developer, I know this is not possible unless the "child" app is installed as an app, supports a URL scheme, the "parent" app knows about, and their interaction can be limited to launching each-other back and forth. 

Note: the situation with jailbroken devices and in house applications is different, but I assume you don't care about those.