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How to delete old files in GIT while keeping history?

I'm far from an expert GIT user.  I'm usually the only one working in a repository at a time, and know how to do the basics like committing snapshots.  I still have to look up how to create and merge branches on the relatively rare occasions I need to use them.

We have some repositories that have gotten large, past the limit of the free hosting service we are using.  Early on, some large files that don't really need to be tracked (like .EXEs) were accidentally included in the files GIT tracks.  There are also a lot of old versions of some large files we'll never get back to.  It would be nice to delete both these kinds of files, but still keep the commit history.

Of course we could copy the GIT repository, clean it out properly, make sure only the files we really need to be tracked are tracked, create a whole new repository and delete the old one (after archiving it on long-term media, of course).  However, that looses the history being easily accessible in one place.

Is there a way to effectively fully delete files with all their versions from a GIT repository while still retaining the history and old versions of all other files?