Hello, our name is Equality 7-2521! Whoops, we really shouldn't have told you that…
In the opening pages of Anthem, Equality 7-2521 introduces himself, tells us about his past, and gives us the essential details about the horrific society he lives in. The poor guy seems to "commit transgressions" a whole lot. The first thing he tells us is that he's committed a sin just by writing.
It's a dark and mysterious tunnel to the past
When Equality 7-2521 finds the tunnel, the story gets rolling. The tunnel makes him commit his first serious – and secret – transgression against society. The tunnel is also what sets off his curiosity about the Unmentionable Times. It's only because he has the tunnel that he's able to conduct scientific experiments and do all the other illegal things he does. (One note about the conflict. Technically Equality 7-2521 found the tunnel in the "past," so far as the story is concerned, he's already been going down for a couple of years when the story begins. But just because it's not in the "present" of the story doesn't change the fact that it's the point that sets the plot in motion.)
Let there be light!
Equality 7-2521 invents an electric light, and this makes things a lot more complicated. Now he feels an obligation to share his discovery with everyone else, which means going public about all the laws he's broken. Then, to make matters worse, he gets so excited about his discovery that that same night he gets caught and whipped and put in prison and all that nasty stuff.
So, how many Scholars does it take to change a light bulb?
Equality 7-2521's confrontation with Collective 0-0009 over his invention (electric light) is the moment of highest tension in the story. The fate of his invention, and his future in society hangs in the balance. When Equality 7-2521's gift is rejected, it turns Equality 7-2521 against his society. It's now a given that he'll leave it for good, and take his light bulb with him. However, we should also mention here that there's kind of a second climax in the denouement…check that out below.
Free to frolic in the forest…
There's no suspense per se once Equality 7-2521 leaves his City for good. He knows no one will follow him into the forest, so he's not threatened anymore. The one thing we really want to know is whether things are finished with Liberty 5-3000, and then she shows up. Then the only question that remains to be answered is what our pair of lovers will do for the rest of their lives.
Finding a house, and the first-person singular
Equality 7-2521 and Liberty 5-3000's discovery the house from the Unmentionable Times brings the action of the story to a close. They literally settle down and start having babies, and Equality 7-2521 starts plowing through the library and then decides to write his little "Why we should all worship our egos" pamphlet (the last two chapters of the book).
In another sense, however, the climax of the book is actually to be found in the denouement after the action has stopped happening. Equality 7-2521's discovery of "I" is what the two preceding chapters have been building towards, and in a way what the whole book has been building towards. It's not the climax of the plot, in the sense of the "action," but you can call it the "internal" climax, because it's the climax of Equality 7-2521's self-development.
If you want to, you can also reread the other stages in light of this climax: the conflict becomes the moment when Equality 7-2521 tells us about the Unspeakable Word, and the complication becomes…well, we'll let you pick that one.
"EGO" is the last word of the book, and yes, it is written in all capital letters. This one word sums up Anthem coherently and concisely.