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.
Tongbu App Store. It may be one of the best ones but it may have some problems. It crashes some times but can easily be fixed by connecting to computer.