Petroff and Caro-Kann come to mind. The Steinitz defense of the Ruy Lopez is not bad either (1.e4 e5 2.Nf3 Nc6 3.Bb5 d6)
Best equalizing defense against e4

I disagree about the Steinitz, I think white can fight for the advantage, (although I am no expert on the Ruy Lopez. The Petroff equalizes

Caro-Kann, Ruy Lopez Berlin
Of course when I play the Caro I play to win. Everybody knows that. I don't play to win in the opening though.
While the Petroff's is a great option against 2. Nf3, black would still be on the hook for replies against such moves as 2. f4, 2. d4 and 2. Nc3, not to mention the assorted intricacies of the Four Knights opening. I could vote for the Caro-Kann or even something like the Alekhine, which also offers great practical chances.

In my opinion I would either say Petroff's or Caro-Kann, both have a very good draw result and if played correctly can prevent white from gaining any crushing attacks.

While the Petroff's is a great option against 2. Nf3, black would still be on the hook for replies against such moves as 2. f4, 2. d4 and 2. Nc3, not to mention the assorted intricacies of the Four Knights opening. I could vote for the Caro-Kann or even something like the Alekhine, which also offers great practical chances.
Of course, the Alekhine is all about counterplay from move 1.

in the caro kann they can play the panov though and that tries for an advantage. . .
In other threads, people love to talk about what they think the other player wants to play. So I will do that here. :-)
When white plays e4 they didn't play d4. The Panov-Botvinnik attack against the Caro-Kann leads to d4-style positions. Indeed, it can directly transpose. The following lead to the same position.
1.d4 d5 2.c4 e6 3.Nc3 c5 4.e3 cxd4 5.exd4 Nf6
1.e4 c6 2.d4 d5 3.exd5 cxd5 4.c4 Nf6 5.Nc3 e6
I don't dispute that the Panov is a great choice; in fact it is what I would play against the Caro-Kann, if I were ever to play 1.e4. But it is still one of many great choices.

i'll cast a vote for the petroff defense, as the opening that the most e4 players would not want to face. as one extreme example, world champion anand switched to 1.d4 for his wc match with kramnik-- and it was no surprise to anyone!
i think vs the caro white has at least two ways to get a nice edge-- probably more.

I thought that Anand was avoiding the Berlin, not the Petroff.
Uh oh David, what are those ways? Does this explain why it's not popular at the top level recently? In the good old days, even Tal played the Caro-Kann against Botvinnik.

first and foremost the simple nc3 de ne4. this is the main reason. secondly panov is pleasant for white, but i don't think that's a reason to avoid it-- it's about equivalent to various other openings that are a bit pleasant for white, as you noted with the transpositions.
what i don't know as much about right now is the advance variation, but i suspect that's the third way for white to get an advantage. and i think some more ppl will be taking that up as white in the future.
maybe anand was avoiding both? i thought petroff... kramnik's played it a lot more consistently.

first and foremost the simple nc3 de ne4. this is the main reason. secondly panov is pleasant for white, but i don't think that's a reason to avoid it-- it's about equivalent to various other openings that are a bit pleasant for white, as you noted with the transpositions.
what i don't know as much about right now is the advance variation, but i suspect that's the third way for white to get an advantage. and i think some more ppl will be taking that up as white in the future.
maybe anand was avoiding both? i thought petroff... kramnik's played it a lot more consistently.
the capablanca variation gets an edge for white? Seems pretty equal to me.

i don't know what the capablanca variations is, i confess. but as far as i know 3.ed and 3.nc3 give white an edge, and in particular 3.nc3 is not so much fun to play against as black.

i don't know what the capablanca variations is, i confess. but as far as i know 3.ed and 3.nc3 give white an edge, and in particular 3.nc3 is not so much fun to play against as black.
the capablanca variation is just the main line Nc3 like you said. I play the caro-kann and it is my favorite to play against because I have studied it the most compared to the panov or something like that. It might give white a slight edge at the GM level, but it seems equal to me. ;)
What is the best non-aggressive opening in your opinion against e4 that just gains equality without trying for any counterplay.