There are actually some lossless operations you can perform on a JPG that still won’t cause any data loss. They aren’t very powerful – rotation (landscape -> portrait orientation) and L-R or Up-Down flip are the ones that come to mind – but in most (sensible) programs saving these changes cause no data loss. If you think about it, you can see why – there’s no actual change occurring in the pixels, just rearrangement of the layout pattern of the pixels themselves. Oh, and editing EXIF data (that records details such as what camera took the shot, what ISO, shutter speed, aperture, etc) shouldn’t cause data loss (again, for obvious reasons).

I can’t of course guarantee that your favorite image viewer or editor handles these operations, but it shouldn’t. If you want to check for sure, take a large JPG, perform one or more of the above operations, and resave under a new name. The files should have the exact same byte size, unless you changed the amount of data in the EXIF tags (for instance, Canon -> Nikon = 5 chars each; Canon -> ABCDEFGHIJ = +5 characters, or 5-10 bytes increase in file size).