Rand wants to kill the DO for good. Will he manage that unlikely feat? I believe the answer is "NO".
First, a closely related question is whether or not time will remain circular after the LB. BS said the following on the subject.
Interview: Jul, 2009 BarnesandNoble.com Book Club Q&A with Brandon Sanderson
The question that I have for you is, now that you know the ending of Wheel of Time, after the final book has been released will it be a world that you could set a game in? Or will it be like Tolkien where after the end of Lord of the Rings the world is pretty much over? I ask because it looks to be a great place to set an RPG and I want to know if I should be looking to a time before The Eye of the World or if I should run a new age?
I'm going to stick pretty close to things Mr. Jordan has said or implied regarding this. Things he has said have implied strongly that it is not going to be like Tolkien; though the Wheel will eventually turn to a point where the One Power is forgotten and the land becomes like our world, that is NOT the Fourth Age. I think it would still be a fantastic place to set an RPG game.
If the Wheel of Time remains a wheel the time will be cyclical. That's what cyclical time means.RJ had some similar quotes too. E.g. this one
Interview: Jan 25th, 2005 TOR Questions of the Week Part II (Verbatim)
Week 18 Question
Who were the first channelers, and how did they learn? By trial and error? Are there any Ages where channeling does not exist?
The first people to discover the ability to channel learned through trial and error, with fairly high casualty rates until they learned enough not to kill themselves accidentally. Their appearance marked the beginning of the previous Age to that of the books, or at least the end of the Age before that one.Yes, as I have set things up, there are Ages when no one has any idea of how to channel or even that the One Power exists. Our own, for one. (The Wheel of Time turns.)
This is pretty definitive about time remaining circular IMO. Time remaining circular is often linked to the DO surviving in the books. The biggest clue to both is actually given in the prologue of book 1 of the series:
And it came to pass in those days, as it had come before and would come again, that the Dark lay heavy on the land and weighed down the hearts of men, and the green things failed, and hope died. And men cried out to the Creator, saying, O Light of the Heavens, Light of the World, let the Promised One be born of the mountain, according to the prophecies, as he was in ages past and will be in ages to come. Let the Prince of the Morning sing to the land that green things will grow and the valleys give forth lambs. Let the arm of the Lord of the Dawn shelter us from the Dark, and the great sword of justice defend us. Let the Dragon ride again on the winds of time.
—from Charal Drianaan to Calamon,The Cycle of the Dragon.
Author unknown, the Fourth Age
The first bolded part reconfirms that the time will remain circular but it also says that the Dragon will be needed to fight the Dark in the future. Certainly, if the DO was permanently killed people would know about it after the LB was over (the quote is from the fourth age) and would not say this.
Also, the second bolded part says that the Dragon will be repeated reborn in the future. The function of the Dragon is to fight the DO. He is never born for any other reason. RJ said that Rand's soul can be reborn as a regular person and need not come as a Dragon every time he is reborn.
This passage is repeated for a good measure at the end of TGH. (BTW, it also says that Rand will bring the Song back and it might play a role in the LB but that is a different subject altogether.)
The header of TGH points very strongly in that direction too.
"And it shall come to pass that what men made shall be shattered, and the Shadow shall lie across the Pattern of the Age, and the Dark One shall once more lay his hand upon the world of man. Women shall weep and men quail as the nations of the earth are rent like rotting cloth. Neither shall anything stand nor abide...Yet one shall be born to face the Shadow, born once more as he was born before and shall be born again, time without end. The Dragon shall be Reborn, and there shall be wailing and gnashing of teeth at his rebirth. In sackcloth and ashes shall he clothe the people, and he shall break the world again by his coming, tearing apart all ties that bind. Like the unfettered dawn shall he blind us, and burn us, yet shall the Dragon Reborn confront the Shadow at the Last Battle, and his blood shall give us the Light. Let tears flow, O ye people of the world. Weep for your salvation.
—from The Karaethon Cycle: The Prophecies of the Dragon, as translated by Ellaine Marise'idin Alshinn, Chief Librarian at the Court of Arafel, in the Year of Grace 231of the New Era, the Third Age"
That's a prophecy so it's got to hold unless the DO breaks the wheel which we know won't happen. The bolded part also strongly couples the repeated rebirths (including the future ones) of the Dragon with fighting the Shadow.
For my money the above quotes are sufficient to settle the issue but there are a few other clues that I find interesting even though they are much less definitive than the ones I cited earlier.
First, there is a note from Herid Fel to Rand
Belief and order give strength. Have to clear rubble before you can build. Will explain when see you next. Do not bring girl. Too pretty.
--LoC, ch 50
This note is clearly of rather large importance in the books. It's also beyond doubt that the part "Have to clear rubble before you can build" speaks about the necessity of breaking the existing seals before building new seals (or whatever will take their place). If the DO was going to be killed this would not be really necessary or of any great importance. Furthermore, Herid Fel expressed rather firm opinion that it's beyond Rand's capacity to seal the Bore perfectly.
“What? Yes, exactly the point. It can’t be the Last Battle. Even if the Dragon Reborn seals the Dark One’s prison again as well as the Creator made it. Which I don’t think he can do.” He leaned forward and lowered his voice conspiratorially. “He isn’t the Creator, you know, whatever they say in the streets. Still, it has to be sealed up again by somebody. The Wheel, you see.”
--LoC, ch 18
If this is beyond the Dragon's abilities, killing the DO should be quite out of the question. Now, of course, Herid Fel may be mistaken about this but I don't think he is. I has a strong feeling that he is speaking with the voice of the Creator (I mean RJ of course) about all these matters. LTT thought much the same thing in the last moments of his life
Because in his pride he had believed that men could match the Creator, could mend what the Creator had made and they had broken. In his pride he had believed.
I think the references from tEoTW prologue are particularly important because RJ said repeatedly that he knew how the series was going to end from the very start and tailored tEoTW prologue with the ending in mind.
Next, there is a constant theme in the series that there has to be a balance between the good and the bad. This is repeatedly mentioned when discussing Rand's ta'veren nature. It's mentioned in other situations too. Thom spells it out in TDR in a conversation with Mat.
"There’s always a balance, you know. Good and evil. Light and Shadow."
--TDR, CH 31
In ToM the same idea is further accentuated when Gawyn discusses bloodknives with Kaisea:
Kaisea laughed again. "Weaknesses? Great Lord, did I not say that they are the finest warriors in Seanchan, enhanced and aided by the Empress's favor, may she live forever?
"Fine. What about the ter'angreal, then? It helps the assassin when he is in shadow? How can I stop it from working? Perhaps light a large number of torches?"
"You cannot have light without shadow, great Lord, the woman said. "Create more light, and you will create more shadows."
--TOM, ch 33
Moridin believes this too.
"There is a way to win, Moridin," Rand said. "I mean to kill him. Slay the Dark One. Let the Wheel turn without his constant taint."
Moridin gave no reaction. He was still staring at the flames. "We are connected," Moridin finally said. "That is how you came here, I suspect, though I do not understand our bond myself. I doubt you can understand the magnitude of the stupidity in your statement."
--tGS, ch 15
Moridin doesn't have to be right about this of course but it's quite clear that he is not lying here and actually believes what he says. And Rand is acting fairly dumb in this scene. He totally misses another useful (and true!) fact that Moridin tells him right before which is that their link was the reason he managed to get into Moridin's dream. I believe that Moridin is correct on both counts here and Rand will eventually find that out, likely painfully, in both cases. What might happen is that Rand will try to kill the DO and fail. Killing the DO seems to be his plan A at the moment. It would certainly be preferable to simply sealing the DO away.
Here is how it may possibly happen. There is a good reason to believe that Rand will have to go to Shayol Ghul twice.
It has been long noticed that the prophecy "He shall hold a blade of light in his hands, and the three shall be one" means that Rand will get himself a second hand at some point.
On the other hand, on the AMOL cover we see Rand, Moiraine and Nynaeve going to the Pit of Doom. His left hand is behind his back but the artist confirmed that this is just to hide the stump. If that info is correct it means that Rand won't have his left hand back by that point. The only explanation I can see for this is that the AMOL cover shows Rand's first trip to Shayol Ghul, later he gets his left hand back (say, by dying and being pushed out of TAR by Nyn), then goes to Shayol Ghul the second time and seals the Bore.
Another clue in this direction noticed by Suttree on the forums is the following quote from TSR
“Stilling is not a thing anyone would choose to study, you understand,” Nynaeve continued. “It is generally accepted to be irreversible. What makes a woman able to channel cannot be replaced once it is removed, any more than a hand that has been cut off can be Healed back into existence.” At least, no one had ever been able to Heal stilling.
--TSR, ch 5
This is a typical foreshadowing. We know that one half of it has been fulfilled by Nynaeve herself and the other one is likely on the way. This ties in nicely with references to Nynaeve healing a man dead three days.
Aes Sedai had been stilled studying ter’angreal. Burned out, it was called when it happened by accident, yet it was just as final. Nynaeve did not think so, of course, but Nynaeve would not be satisfied till she Healed somebody three days dead.
--LoC, ch 7
Nynaeve pushing Rand out of TAR would neatly fulfill both things.
So what would Rand be doing at Shayol Ghul on his first trip there? We can assume with reasonable certainty that he is going to break the seals there. That's his firm intention and the line of the second dark prophecy "and the First Among Verminlifts his hand to bring freedom to Him who will Destroy" supports this too, provided that the First Among Vermin means Rand, of course. But this can't be all he wants to do at SG. He can break the seals anywhere, no need to go to Shayol Ghul for that. The seals are very brittle now and are easily breakable.
As I said, Rand's plan A seems to be to go to SG, break the seals and kill the DO. I believe that's what he'll try to do. He'll succeed in the former but not the latter. The attempt to kill the DO should exhaust Rand quite a bit. It should be a much harder affair than Maradon and he was very tired after that. This should then make him vulnerable and then there will be counterstrike by one of the Dark Siders which should eventually lead to Rand's death. He even hinted that something like that might happen after the Maradon fight.
Rand shook his head. "I am not to fight this war, Bashere. Today's battle exhausted me beyond what I should have allowed. If my enemies were to come upon me now, I'd be finished.
--ToM, ch 32
Lastly, let me mention that I used to think that there was an (admittedly very remote) possibility that Fain will replace the DO as the primary source of evil. But a couple of quotes by BS pretty much rule that out.
Interview: Dec 25th, 2010 Stormblessed.com Interview with Brandon Sanderson (Verbatim)
Everyone's favorite Wheel of Time question has been answered—after Towers of Midnight, we now know who killed Asmodean. It doesn't seem right to leave you without a WoT question, so in honor of Asmodean: What do you think would happen if Rand managed to hurl Padan Fain through the Bore into the Dark One's prison?
The Dark One would spit him back out because he tastes bad.
This pretty much says that Fain is nowhere up to snuff to kill the DO despite what he himself may think. And another quote
Interview: Dec 8th, 2007 Dragonmount Interview with Brandon Sanderson (Verbatim)
And now, just for fun:Who's nastier: Moridin, or Padan Fain?
No contest: Moridin. You can't really top Ishamael when it comes to nastiness, particularly if his opponent is just a little Darkfriend weasel. Yeah, Padan Fain facilitated the attack on Emond's Field and all that, but he's still just a weasel.
This is from before tGS but it still shows what BS thinks of Fain as a character and of his power relative other characters. BS had access to the notes by then so if Fain was going to have a really grand role like killing or replacing the DO, BS would never say this. Incidentally, IMO this quote also pretty much guarantees that Ishy will not turn back to the Light (not that I ever for a moment believed that he would).
In summary, the DO won't be killed or replaced and the cyclical time will continue as before. It is possible (and even likely IMO), however, that the sealing of the Bore this time around will be somehow special. I don't know how but I do expect there to be a twist of some sort.
Update: I believe memory #24 might be relevant to the above discussion.
"There were many good years. Good decades, good centuries. We believed we were living in paradise. Perhaps that was our downfall. We wanted our lives to be perfect, so we ignored imperfections. Problems were magnified through inattention, and war might have become inevitable if the Bore hadn't ever been made."
The speaker is obviously Rand. None of the Forsaken would say anything of the sort. Since this much is immediately obvious, why is this memory being given to us? What is there to discuss? I think it's about the content of the above passage, rather than of who the speaker is. It looks like boring philosophizing at first glance but perhaps there is more to it than that. Rand is talking about the necessity of balance between the Light and the Darkness, a recurrent theme in WoT which I highlighted earlier. What he is saying is that even if the DO were not around the Dark would come eventually. This may be related to the idea that even if the DO is killed or permanently removed from the Pattern there will be a need for the Dragon in the future. The main quote that I think proves that the DO will survive (from tEoTW prologue) rather carefully says that the Dark will come again many times, not the Dark One. I don't much like this as I don't find the idea of a superhuman messianic figure as a Dragon being necessary to fight regular human created evil. However, it does agree with the cosmology of WoT where ta'veren are needed from time to time to fix problems in the Pattern and it might be necessary to bring out the big gun (the Dragon) once in a long while to fix major problems.
I still don't believe the DO will be killed however. I think this memory is more about the fact that even if Rand seals the Bore as well as possible the Dragon will still be needed in the future even if the Bore remains closed.