I still sometimes have the "we can't quite tell if you're a human" popup happen when I try to do stuff, with attached captcha. I think there should be a privilege that comes with a certain amount of rep that stops that from happening, because it's kind of annoying and it's rather unlikely someone with more than a certain amount of rep is a bot. I'm not seeing it frequently enough that I'm interested in altering my software or network to obtain a work-around.
After reading the posts linked by @doppelgreener, I think this should be done even more. It seems silly that you have to have 10K rep before the system decides you probably aren't a bot. In fact, it seems counterproductive as a 10K user who is a bot (because their account got hacked) can actually cause harm to the site (as a moderator) while the rest of us couldn't actually do much besides screw our account over. Spam bots are target at sites with low entry barriers, which SE is not. Even a measly 5 rep requirement would probably be fine.
Additionally, why do we care if bots use our site? If they're contributing high-quality questions and answers (which a 10K mod would have to be) does it matter? I understand that a botnet of coordinated upvoting and downvoting could be a problem, but don't you need a baseline amount of rep to upvote and downvote? And isn't there a 200 rep single day maximum?
I think the system is more than safe enough without a captcha on anything but account creation, but I'm not a computer security expert by any means and I can see how it might be and issue so I'll leave that alone. I really can't see how letting users with more than, say, 100 rep do stuff without asking if they're human is a problem. And according to the comments, users with 10,000 rep (which is supposed to give some anti-captcha benefit) still see the popup occasionally.