I should admit I don't completely understand why reputation seriously bothers some people. I think (and I think others share this view) that reputation is just there to make the system work (analogous to funding money for research) and provide some feed back, but other than it does not really mean much (though I have seen it causing some participation addiction). I personally found statistics like average total votes for answers more interesting.
I think if we have a better understanding of the reason some dislike reputation, we may find solutions that work for them without altering the system.
First, let me point out that even if there was a reputation opt-out it would still be possible for others to calculate how much reputation you have (and it is not really difficult using the SE API).
If the issue is that seeing your or other users' reputation bothers or distracts you then you can use userscripts to hide them.
If the issue is that you don't want to have the privileges that come with reputation then you can donate your reputation on questions you like. This is particularly fun with the new bounty system which allows you to select the reason for your bounty. For example, you can put a bounty to award it to some already existing extraordinary answer.
After reading this discussion over tea I think I partly understand the motivation.
"I answer questions under random user names so people actually have to engage with the mathematical content rather than simply vote based on name recognition."
However, it seems to me it is very difficult to have the following two at the same time:
- retain the ownership of posts,
- not have any publicly accessible history of behavior.
I think the issue is not really reputation, at least not in this case, but not having any public history so the posts are judged purely based on their content. Users who don't care about 1 already can post completely anonymously.
I think we can request a feature where the information about the owner of a post is not publicly accessible (but it will be to moderators and maybe also to trusted users). I think the reason has merits and such a feature would be interesting. I don't know if SE would find it a reasonable feature-request, but it might worth discussing/asking for such a feature.
A negative side of posting anonymously would be lack of meaningful interpersonal interaction, I mean think about a math conference where people are wearing masks to remain anonymous to keep the focus on math. I don't think I would find it a pleasant environment.