The Three Destroyers

Perhaps Lews Therin (whatever part of him is left to occupy Rand's mind, that is) has finally given up hope. Maybe it was in the time his soul spent between dying and being incarnated as Rand (wherever it was, Tel'aran'rhiod or otherwise) he realized the pointlessness of his struggle against the Shadow. He knows he is doomed to repeat these battles as long as the wheel turns and wishes to end it all. It might explain that strange desire to break the seals, but it could also explain who the third destroyer is.

You can see how Lews Therin might possibly feel about the One Power, the force that drives the Wheel, and anyone who channels it. He might feel as though channeling could only bring about more suffering, making all channelers destroyers. Wasn't it Aes Sedai, wielding the One Power who freed the Dark One? Didn't they (the men anyway) break the world? Perhaps allowing the Dark One to win is the only way out. And Lews Therin is crazy enough to believe that.

I propose that the "other one" Lews Therin mentions is Alanna. At that point she is the only person in Rand's head besides Lews Therin, and is also a channeler. We all know Lews Therin is uncomfortable around Aes Sedai, but he was terrified of Alanna and her presence in Rand's mind after the bonding. It's a simple explanation that (I think) fits best. Lews Therin has grumbled about Alanna before (sorry if I can't remember when or what was said.) I rule out any of the Forsaken only because Lews Therin usually speaks of them as a group, or individually by name. Other possibilities include Taim and Cadsuane, the two other major channelers on Rand's mind now. Verin doesn't bother Rand as much as these two and a single Asha'man or Aes Sedai probably wouldn't stand out enough for Lews Therin to mention him or her as the "the other one." Rand's only contact with Moridin was in Shadar Logoth (if it even was Moridin.) I doubt Rand remembers the incident clearly but perhaps Lews Therin does remember and knows something Rand does not. So there's the fourth possibility.

However I still feel that Alanna is the best candidate for the third destroyer because she is a constant presence in Rand's mind (whose existence troubles or frightens Lews Therin.) She fits the bill of a channeler as a destroyer; she bonded Rand without his consent (essentially raping him) which is doubly a crime to Lews Therin, with his particular feelings toward channelers.

Pento McGreno says: Hmmm...let me think...no. But feel free to send a few more like that, they're always good for a laugh.