What is this person gibber-jabbering???
A persons theme/site will auto-size the image according to where it's needed as it should. Although, your settings do have an "exact size" setting, so that I'm guessing should work for this person.
But, really if this person want's a particular size at a particular place in their site, that place needs to designate the size. You can't have different sizes of an image unless you create different uploads and then the place where you show it calls that particular upload only, and not the standard avatar.
In other words.... If they want to have an image that's not resized, then they need to create it separately and upload it separately, and call it separately. The complexity this creates is negligible to a site having one image that it resizes everywhere in it according to the sites needs.
Plus, use a "Cache" plugin like Total Cache as you should be.... if you're so worried about resource usage.
One other thing, I know wordpress itself creates copy's of an image at the various needed sizes. Oh and guess what.... I just checked the UPLOADS folder, and I see the AVATARS ARE ALSO RESIZED....
In other words, if your theme location is working like it should, it should be using a smaller image sized copy. It's not actually re-sizing on the fly. Check your upload folder yourself.