With large pretrained models, existing methods have effectively improved instruction-based video editing. However, most of them rely on an in-place editing assumption. They align the edited video with the given source clip frame by frame over a fixed time span. This pattern fails for open-ended streams, e.g., restyling a live game or applying a camera move to an ongoing shot. In such cases, edits must extend to future frames as they arrive, rather than be applied to a static input clip. In this paper, we study this setting and name it infinite video editing: given a preceding segment and an edit request, a model must generate the next segment that continues the stream while applying the requested edit. This process repeats as an unbounded sequence of edit instructions arrives. This task brings two challenges: the edit must be a faithful continuation rather than a frame-wise rewrite, and generation quality must remain stable as edits accumulate. To address them, we first design a data-collection pipeline for infinite video editing. Based on the collected data, we propose InfinityEdit, a lightweight edit adapter that equips a streaming video generator with unbounded editing ability. The adapter contains three attention modules. History cross-attention guides the denoising frames using the input frames. Temporal causal self-attention keeps temporal cues flowing only from earlier frames to later ones. Edit cross-attention injects the edit request into generation. During inference, the adapter is activated only in the chunk where an edit request arrives. Subsequent chunks are generated by the original model with a reset anchor frame. This scheme applies the edit while preserving the original model's infinite generation ability. Extensive experiments show that InfinityEdit faithfully continues the stream under each edit, and stays stable over unbounded edit sequences.