It does not exist. You can specifically Share a video with Circles.
> influence your search results in ways you don't want,
In the upper right on your Search page, you can click "Hide personal results."
> or show up awkwardly in other strange ways while using other Google products like Gmail
[citation needed]
> The real problem is the effects are unknown and upvoting is no longer a contained action with a specific social purpose.
What if you want to share with your circles, what if you want things you've +1'd to influence your search results, what if you want to see better ads rather than crappy ads?
> my boss reading that I "liked" a LOLcats video while at work
Don't Share that with your boss. That's what Circles are for.
> my wife seeing that I "plussed" a review of Avengers by a woman in lingerie or something
+1's are inherently public.
I can appreciate that you like to give feedback to content creators saying, "Hey, I liked this!" Google is tying the "I" in that sentence to your Google+ account.
I think the days of "Anonymous Likes" are somewhat over... They were always susceptible to spamming, anyway. And they were never really as anonymous as we liked to think they were.