Library OS (LibOS) architectures implement the OS personality as a user-mode library, giving each application the flexibility to choose its LibOS. This approach is appealing for many reasons, not least the ability to extend or customise the LibOS. Recent work with Drawbridge showed that an existing commodity OS (Windows 7) could be refactored to produce a LibOS while retaining application compatibility. This paper presents Bascule, an architecture for LibOS extensions based on Drawbridge. Rather than relying on the application developer to customise a LibOS, Bascule allows OS-independent extensions to be attached at runtime. Extensions interpose on a narrow binary interface of primitive OS abstractions, such as files and virtual memory. Thus, they are independent of both guest and host OS, and composable at runtime. Since an extension runs in the same process as an application and its LibOS, it is safe and efficient. Bascule demonstrates extension reuse across diverse guest LibOSes (Windows and Linux) and host OSes (Windows and Barrelfish). Current extensions include file system translation, checkpointing, and architecture adaptation.