Are I right in thinking your saving in between steps here?
No. I didn't save at all. After each of the operations I used File, Save As, and the dialogue box asked me what Quality to use, and indicated what the size would be for the Quality setting I selected. Having noted what the size would be I then cancelled the dialogue, and used PS History to take the image back to its as-opened state to repeat the process for the next operation.
If so thats not a good way to work, although the jpeg is getting bigger file size wise, the recompression every time it's saved will reduce the quality and bring in jpeg artifacts eventually.
If the OP need to revist his image it would be better to save as a tiff for the in between stages and output as a jpeg at the end of the processing.
Although not relevant in this case, this is all certainly true and good for the OP to know about.
Mind you, having said that, from time to time I decide that a PP'd image (saved at JPEG 10) would benefit from a little tweak, and rather than redo from scratch with the original (which for some of my images can be a non-trivial exercise), I usually load the PP'd image, do the tweak and re-save as JPEG 10, which is quick and easy.
Being concerned about the issue of progressive quality loss with repeated saves, some time ago I did several very close comparisons of first-saved and second-saved versions, and as I recall I did not manage to convince myself that I could see any IQ loss. I believe there must have been some loss of IQ, but it was below my perceptual threshold.
I haven't done that sort of minute comparison since then, but I do always do a first-save versus second-save (and occasionally third-save) comparison to see which version I prefer, and sometimes this is a rather detailed comparison. Here too, I don't recall noticing any IQ loss. This may not apply generally though - I suspect that different types of image may react differently in terms of how easily they develop artefacts.
BTW, and FWIW, I use a comparison method that I believe is quite revealing of differences. I arrange it so that I can flick back and forth between versions with one version exactly replacing the other in terms of size and position. I keep my eyes fixed on a particular part of the image and watch how that area changes as I flick back and forth, for example by rocking the mouse wheel back and forth, which allows me to not take my eyes off the image. I find that differences between versions really jump out at me when I do this.