Just to clear things up, this is called "User Flair", and exists on the StackExchange network data for a user under the 'flair' tab, or accessible by specific URLs such as:

    https://askubuntu.com/users/flair/10616.png

(that one's mine)

User Flair such as that is not customizeable beyond the theme in use according to all the Meta.SE threads I checked for this.

Known themes to exist are (appended by `?theme=` to the PNG URL):

 - None (the default)
 - `dark` - makes a dark theme
 - `clean` - can't tell the difference from default but it works
 - `hotdog` - makes a lot of red.

I don't think you can customize this to have anything such as "latest answer" or "most voted questions" - at least, there isn't any such 'flair' that I have found for this.  I went hunting for such things and didn't find them.  If they exist, they aren't documented, though I highly doubt there's any type of 'custom' embed-able code that gets you the other stats.

(There may have been at one point, that was done via a third party service, but not anymore.)

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And just to confirm I made inquiries via certain discreet channels, and got a "Yeah there's nothing other than user rep you can put there" as the reply.