How do I know I reached a file's maximum size when using ofstream?

First off, maximum file size is a filesystem limit, not an Operating System limit. It will even vary for a particular filesystem, based on how the filesystem was formatted.

As for how you'd figure out that you'd reached the limit, your code will likely throw an exception when that happens, which you'll then be able to relate back to the OS error codes.

Note that the actual limit is pretty darn big for a "typical" EXT2 filesystem - in the Terabytes. You'll likely never reach it, in practice. If you seriously are accumulating Terabytes of data, you might want to consider whether there's a more reasonable way to store it, rather than a single gigantic file.