Thinking on this question, I thought it would be a hard one to answer, as I rank all three films very highly. In the end, however, it was relatively easy, and the answer is The Two Towers.
I preface with three crucial points;
Firstly, this is relative only to the extended versions of the films. In my mind, the theatrical versions don't even exist, and I will never watch them.
Secondly, my judgements here are by fractions of degrees. All three are exceptional films that I dearly love.
Finally, I approach these films as their own work, independent of their source material. For me, deviation from the books isn't necessarily a mark against the film. Film and books are two distinctly different mediums, and what works in one may not work in another. I simply assess the film that is presented to me.
Fellowship is a masterful film, and had the most impact on me originally. Of all three films, I saw this one the most number of times in the cinema. I went back again and again, not able to get enough of it. But in wider context, great as this film is, it suffers for being a "set up" film. Much of its time is spent paving the way for the following films. While Fellowship is actually the shortest of the three films, it often feels the longest to me. In the end, for me it lacks emotional impact. While there are moments, they are just that; moments, isolated by long periods of set-up and travel.
Towers wins for me because of its tone, and its balance between spectacle and drama. It has its flaws - none of the films are perfect - and in places the filmmakers over step (particularly in relation to Gimli and Legolas) but the film exudes a powerful pathos; the story reaches its lowest most desperate point during this film, and with it comes the greatest drama. For me, many of the most powerful scenes in the entire trilogy occur in this film; particularly revolving around the battle of Helm's Deep.
There's spectacle on a fantastic scale, as the stakes are raised, but it never raises beyond the unbelievable. I think a big part of this is that the smaller scale of Helm's Deep enabled the battle to mostly be shot live. The latter battles of Minas Tirith and the Black Gate, by their nature, were assembled for a mix of isolated heavily CGI'd shots, and I think you can really tell. Helm's Deep was shot on a full size fortress set, in the dark, in the rain, between two armies of hundreds and hundreds of people, over three grueling months.
For me, with the Rohirrim facing the annihilation of their people, the film best captures what's at stake in this narrative, and what will be lost if good lose.
Meanwhile, on the other side of the spectrum, the final aspect which settles this film as the superior film for me is Gollum. In the books I found Frodo and Sam's narrative tedious beyond belief, but in the films Gollum is given centre stage, and the desperately tragic tale of his life, and his almost-redemption is incredibly powerful, particularly juxtaposed against the desperate battle at Helm's Deep. I can barely stand to listen to the closing credit music Gollum's Song without tears filling my eyes.
Return is, for me, the weakest of the three, and in testament to how close the three films are, despite being my least favourite, it nonetheless features some of my favourite moments in the whole trilogy. But here drama takes a backseat to spectacle for most of the film. I found the battle of Minas Tirith really disappointing, because the scale became so absurd, and the level of "one upmanship" spectacle became so ridiculous, that I just didn't invest as much in what was happening. For me the most poignant battle sequence in Return is Faramir's failed attempt to recapture Osgiliath. By comparison the Rohirrim cavalry charge (despite great speeches preceding it) felt like a cartoon.
