Before we begin, let's just say if there is one playwright who is going to break the "Classic Plot Analysis," it's probably Samuel Beckett. Still, we've made a crack at fitting Endgame into the mold. Now, the initial situation is that the apocalypse has already happened. Hamm's parents are in their trash bins, where he has put them, Hamm is blind in his wheelchair, and Clov is stuck tending to everyone, not confident enough to leave.
Much of the point of the play is that the initial situation is the end. The end has already happened. The characters have just failed to come to terms with it. So, in that very first scene where Clov draws back the curtains and takes the sheets off of Hamm and the trash bins, the end has already taken place. As Hamm says, "The end is in the beginning and yet you go on" (1.688).
Here's one way of seeing the conflict of the play: will Clov leave Hamm so that the two of them will die separately or will he stay and continue to prolong their mutual suffering? This is a question that is asked repeatedly in the play, and comes to a head every time that Clov threatens to leave and Hamm explains to him Clov can't.
The dialogue between Hamm and Clov determines the main conflict of the play, and also postpones the conclusion. At one point, Hamm tells Clov that what keeps Clov there is the dialogue. Both Hamm and Clov understand that their constant bickering is as much a way of avoiding the underlying conflict as of dealing with it. Neither of them really wants to address their problems head on because there is no possibility of positive resolution. They either die alone or together.
In a way, the revelation of the existence of Hamm's parents in their trash bins is one mechanism to make him a more rounded character, of humanizing him and showing where he came from. Yet, when we see how Hamm treats his parents, he also starts to seem even more of a tyrant than he does with Clov.
In Nagg, we have a vision of what Hamm might be like if he were not in control of the entire situation (i.e., pretty pathetic). Yet, Nagg is also the first character to challenge Hamm's authority, when he wishes that Hamm would be put in a situation where Hamm really needs his father, as he did when he was a small boy. This is a key turning point in the play, and may help clear the way for Clov's later defiance of Hamm.
Now, you probably noticed that this play isn't exactly boiling over with action. If we were going to talk about the climax of action in the play, then we'd probably have to go with Clov dumping a tin of insecticide down his trousers to kill a flea. If we're talking about a thematic climax (which we are), then the exact point is much harder to define. Notice that this is only one of several possible candidates.
We have chosen Hamm's story, however, because this is one of Hamm's most honest moments in the entire play. It is where he comes closest to defining his relationship with Clov, and attempts to trace back the origins of their current situation. It is an unusual moment of vulnerability for Hamm, and it just precedes the point where his father curses him, which makes him even more vulnerable. If the end had already come by the beginning of the play, then this story is the point where Hamm tries to figure out when the beginning was (and therefore, where the end was, too).
Ah, suspense. Another element that is particularly lacking in this play. Tension does arise, however, when characters seem to foreshadow what will or might happen later on in the play. By sheer coincidence (or due to the fact that they all seem to despise each other), this often happens when one character is cursing another. When Hamm curses Clov and imagines him being alone with no one to pity him, he is predicting what will happen to Clov when or if Clov leaves him.
When Nagg curses Hamm, he echoes Hamm's threat to Clov and presents him with the same scenario. He imagines Hamm being alone in the dark and calling out for his father because he really needs him. This is what might happen if Clov actually abandons Hamm, and it is what we get a glimpse of at the end of the play, when Hamm calls for his father. Clov, for his part, threatens to leave Hamm countless times. After a while, the tension of these moments begins to wear off, but it does come back in full force at the end, when Clov actually prepares to leave the house.
Things begin to wind down as Clov and Hamm make their final speeches to one another. Clov, like Hamm with his earlier storytelling, tries to fathom how things began. He thinks back to a time when people made him all sorts of promises about happiness and the possibility of understanding the world. He concludes that he is looking forward to that point when he falls, and it seems that he is imagining that point on his journey where he is all alone and collapses from hunger and exhaustion.
After Clov leaves, Hamm makes a long speech about the nature of ending. He again reaches back to that moment when he obtained Clov from Clov's father, but then, in frustration, casts his possessions away from himself and covers his face with his handkerchief. These are Hamm and Clov's formal preparations for "the end."
Remember how we defined the conflict? Well that's also how we define the conclusion. Will Clov stay or will he go? The conclusion is that brief moment at the end, when Clov stands in the doorway in his Panama hat and just stares at Hamm, unmoving. We do not know if Clov will change his mind or if he will actually leave (for one thing, note that he never set the alarm clock like he said he would).
It is quite possible that every day for the last five years, Clov has dressed and prepared to go, but remained in the house with Hamm. Or perhaps today is the day, and he will actually leave him. The conclusion is ambiguous, but what is especially apparent in this closing scene is how little it matters if Clov stays or leaves. The end happened long ago, and Hamm and Clov are defeated either way.