I did try changing the db_version value. Then it told me I needed to upgrade ... but once I did that, it sent me _right_ back the same old "No upgrade needed" / home page loop. So I don't think that did anything.
What I finally did was move my themes directory, my plugin directory, *AND* several files that W3 Total Cache puts at the root of wp-content/:
At least, it looks to me like these are all part of W3 Total Cache.
Anyway, when I did that, I was able to load the back-end. Then, I was able to re-introduce themes/ and plugins/, and then the W3TC files.
I'll bet you anything that what was the culprit was the W3TC files, because that is what is most anomalous to my install, from a generic install.