Just done a test. I just added a tag to an image that was 6.22Mb. Just the one tag and the file size is down to 5.83Mb. Can only assume that as soon as you hit "Save" further slight compression of the image takes place as it always does when using JPEG's. Also the degree of compression would be the same as the degree used in the first instance. My editor has a range from 1 to 12 so if you chose 9 in the first place this would be the standard for all subsequent "saves". Have just done the same exercise with a PNG image. No change in file size here. Much as I would expect as PNG is not a lossy format. No compression takes place when you save it. So I would conclude that what you are getting is normal for a JPEG! On normal monitors this would make no noticeable difference to the quality, only if you were doing high quality printing to the image maximum size would it notice and then you would probably use a non lossy format anyway.
Answer is, if quality really matters, if you can't shoot the original in a RAW format, use the highest quality JPEG you camera will go to and then save the image to PNG or TIF before anything else and stay in that format until the last minute before you save for the final time, even after adding your titles, tags etc, then save to the highest quality JPEG your editor will go to.