I generally only save nicer stuff. Walnut, White Oak, and anything that is not common. like the aged barn-wood.
Maple, poplar, etc go through in such quantity, that they aren't worth saving.
I do save off-cuts of things with special angles, or shapes until the job is done and installed, makes it easier to repair. Once it is gone/done, I throw stuff out.
The main thing I had trouble with was keeping jigs. I would make something for a specific job, knowing that it was probably a "one-off", but keep it anyway. After a while I would forget I even had it, or where it was if I did.
The "cure" for that was clearly writing the details on it with a Sharpie marker. Job number, date and anything special like router bit diameter and bushing size (if there was one)
I just went through a cleaning a couple of weeks ago, because of an open-house. There was more to throw away than I would have thought.