Asking what happens after the end of Memento is as simple a question as anything about this film. There's the two endings to deal with, but you also have a series of conflicting accounts that make wider truths harder to confirm.
The Narrative End
Quite simply after the end of the film all the events we've already seen play out; Leonard meet's Natalie, attacks Dodd, kills Teddy etc. He's purposely decided to forget that he's reached his prize already and continue the journey. The reason for this lies in many of Leonard's quotes from the film. In comparing himself to Sammy he says "I use habit and routine to make my life possible. Sammy had no drive. No reason to make it work." He recognises, if only subconsciously, that without his crusade there's little point to his existence and that he'll revert to the state he was immediately after the death of his wife.
The Chronological End
What happens after the events of the film, which culminate in Leonard killing Teddy, are rarely addressed in discussion of Memento, but it's fascinating to explore. Obviously there's no Teddy to tell Leonard the truth, meaning the only way for him to even suspect he got him man is the photo he takes of Teddy's body (at the very start of the movie). Proof that his quest is over (assuming he writes something incriminating on it), this would remove that purpose in his life and likely send Leonard in a downward spiral, like Sammy in the home. However, as we know from the narrative ending, he's conditioned to constantly be hunting the second attacker, even after they're dead, so there's a distinct possibility he could somehow restart his quest. Quite what would give him the leap to do that (in the narrative end it was being told explicitly he got his man) is unclear, but it would only take one sliver of doubt (he has removed pages from his police file that don't work with his current theory) for Leonard to willingly forget it. In this case he'd find himself in a loop; one of his permanent facts is that Teddy's car belongs to the killer, meaning he'll be forever hunting a dead man. Which, in a way, makes this the happiest ending; Leonard would be involved in a mystery that he literally can never solve. There's also some extraneous elements that are worth addressing. Dodd and the rest of Jimmy's associates are still out there and Leonard is driving around with $200,000 in the trunk of his car. Too little is known about Jimmy's connections to conclude the impact this could have on Leonard. However, with Teddy, the man who in everyone's minds killed Jimmy, now dead there's a chance all the heat will be called off. Now we've explained the ending, we'd like to present one little theory a little off the beaten track that may further explain just what's going on in the film.
