I don't believe this is a problem.
Firstly, as Bart commented there's only 763 users with the copy editor badge. That's really not many.
Secondly, I honestly believe that one is either a serious editor, or not. If someone really put the effort in to get the copy editor badge, and didn't get edit-banned whilst doing so, I don't believe they'd be able to stop. As evidence there's only 4 users on SO with 500 edits. The numbers climb at a steady rate after that. You get used to editing, begin to like it and continue.
There doesn't appear to be any evidence that this gaming is going on or, if it is occurring, that it's actively harmful to the site. If someone is really so terminally bored with their life that they want to individually add a few thousand back ticks to 500 questions in order to obtain a few pixels then I honestly don't mind. As long as they're positively benefiting the site it doesn't matter that much. Of course, if that is actually what they're doing, they're far more likely to end up edit banned or the subject of a fairly intense meta post. It's just not as easy to do well, or quickly, without bringing large amounts of attention to yourself.
Anyway, I would guess that Fanatic is the easiest badge to get. Look at a few questions once a day; you don't even have to pretend that you care about capitalisation. As evidence 7,929 users have the badge, an order of magnitude more.