Finally I found it. I answer myself:
Clicking the Menu Tools | Process Viewer... opens a separate window which displays the activity log.
Leaving this window open for 10 or 15 minutes, you can see there's a huuuuuuuuuge enormous amount of background things going on in the process.
The look of this window is similar to this:

One of the things I observed is that SourceTree insists in making a "for each" of aaaaaall the repositories that it knows and make real internet connections for all of them. Having 3 or 4 repos could mean something normal, but it detected about 100 repos in my computer.
I'm using SourceTree v3.x but it seems that years ago we had SourceTree v1.x in this computer. The old unused forgotten repos of inactive dead projects are still being recognized by SourceTree (maybe because it's importing some old config file or so) and suddently things that were dead 3 or 4 years ago, came to life.
Not only they came to life, but they came to life every 10 minutes!!!!!
For some reason two of the old repos have some kind of connection to something connected to microsoft. I don't really know the reason, but I really don't mind, they are just dead code. I just want those files to rest in a deep forgotten dir and nothing else.
So the thing was telling SourceTree something like do not do anything I don't explicitly tell you to do AKA leave my old code alone.
And fortunately the possibility is there: Click the menu Tools | Options then select General and navigate to Repo settings.
There you'll find a box named Check default remotes for updates every XX minutes. Uncheck the box.

Boom. Done. 24hrs without the popup. Now the only activity I see is the one related to the repo I'm working on and the rest of the repos are just left alone.