How to clone/fetch a repo getting only the history

We are building ghuser.io (enhanced GitHub profile pages) and any way to get the commit history without files would help us tremendously to scale.

Then you would need to setup a mirror server with GVFS (Git Virtual File System) / VFS For Git support.

Since June 2016 (the OP question) and now (Q4 2018), VFS For Git (since issue 72 is soon to be resolved) has been proposed by Microsoft (Feb. 2017), and allows you to develop with TeraBytes repos(!) without having the files downloaded.

GitHub itself should support it soon.

See more at gvfs.io, although I suspect that a domain name which is now renamed to reflect the new "VFS For Git" name: https://vfsforgit.org.
(Microsoft/VFSForGit.WWW issue 9 is closed, Nov. 28th 2018)

Note: (Feb. 2021), the certificate issue regarding https://vfsforgit.org finally got resolved: see microsoft/VFSForGit issue 1705.

No, or at least, not using any ordinary access. Some sites offer web access, through which you can obtain the contents of every commit object without also obtaining tree and blob objects, but the normal process of receiving objects or thin packs is either truncated at the commit level (via --depth) or is complete.

You can of course see all visible tags with git ls-remote as well as through any sensible web interface (it would be weird to provide something like GitHub's fancy API if you didn't provide the tags that way :-) ).

Note that traversing all commits via a web API may be tremendously slow, either due to having to stop and wait (if you program it synchronously rather than as a streaming process) or due to rate limiting software on the host (GitHub and Bitbucket both seem to do rate limiting).