it often adds a significant amount of extra weight to the file size.
There are a couple of ways that can happen, but if you're going JPEG to JPEG, the sizes will be identical (or at least very nearly so) as long as the quality level from original to the resaved image is the same.
Running this action on a 17.6 MB folder of JPGs resulted in the size almost doubling to 32.8 MB.
Which is exactly what would be expected to happen if the originals were saved with an (example) quality level of 7, and were resaved to 12.
I already knew this would be the outcome, but ran a test anyway. I saved seven random TIFF images to a level 7 JPEG, Baseline (Standard). No matter how many generations I performed the same action to the original level 7 JPEGs, or subsequent copies, the sizes always came out the same. If I saved the level 7 JPEGs to level 12, then they jumped up in size. From 1.6 MB for the level 7 set of images, to 4.1 MB at 12.
The worst part of course being that anytime you save a JPEG as a JPEG you are applying JPEG compression again. Compounding the unavoidable JPEG artifacts in each resaved image. Which you already know, as you stated:
but the whole point is not wanting to affect the amount or quality of the data
The problem is, you can't adjust the DPI without resaving the image. That value is stored in the file itself. The only way to make the change stick is to resave.
Personally, I won't, and have never used JPEG to save my images. Always uncompressed TIFF for flattened images, or PSD for layered finals. Why any Joint Photographic Experts Group users ever decided that a lossy format was a good idea is anyone's guess. You would think the very last file format any professional photographer would use to save their images is one that ruins them by default. I will get a bigger drive to store my data over lossy compression any day of the week.
If you want to save space, use TIFF with Zip or LZW compression. Both will save a good amount of space over an uncompressed TIFF and are both lossless compression schemes.