AGPL is absolutely free software as long as the GPL is considered free software and should not be put into comparison with licenses like the BSL.
The only real limitation on the AGPL compared to the GPL is section 13 which states that you need to make the sources for said AGPL program available to any user who interacts with it. Otherwise the license is practically identical. Otherwise the license is essentially identical to the GPL.
If you diff the licenses, you'll see that other than section 13 and s/GPL/AGPL/g, the actual license terms are identical (if you diff sections 0-12, only the addition of the word Affero to the first line of section 0 will come up). The preamble and afterwords are different but they aren't actually part of the license (only the sections between the lines `TERMS AND CONDITIONS` and `END OF TERMS AND CONDITIONS`). As for the differences in those sections, the preamble provides context for why the license exists compared to the GPL and the afterwords provide a recommendation as to how to provide source access.
So in effect the AGPL is identical to the GPL with the addition of requiring that you make sources available to any user "interacting with it remotely over a computer network". Contrary to popular belief this doesn't mean you need to provide sources for everything that touches it over a network, just that anybody/anything that interacts with it over a network be granted access to the AGPL source (and corresponding sources). The requirement for what is considered part of corresponding sources is exactly identical.
That requirement:
> The "Corresponding Source" for a work in object code form means all the source code needed to generate, install, and (for an executable work) run the object code and to modify the work, including scripts to control those activities. However, it does not include the work's System Libraries, or general-purpose tools or generally available free programs which are used unmodified in performing those activities but which are not part of the work. For example, Corresponding Source includes interface definition files associated with source files for the work, and the source code for shared libraries and dynamically linked subprograms that the work is specifically designed to require, such as by intimate data communication or control flow between those subprograms and other parts of the work.
TLDR: The only practical change in the licenses for a piece of hosted software is that under the GPL the user is the server operator/host and under the AGPL the user is the actual user/client. Whether a given piece of software gets "contaminated" by a AGPL work is no different than if said AGPL work was licensed under the GPL instead.