Wow it is the best story I've read. I love how story is written it toys with the idea that Joker is really immortal but it makes you doubt it to. I mean I genuinely want to believe Joker is who claims in the story but at the same time I have reason to doubt it. Ultimately if I can say one thing I don't want Joker to be a stantard run of the mill bad seed like how that Doctor believed because that would be boring and unoriginal.
If there's one thing I can say is I never want Joker's origin to be boring and him being a child psychopath would be the most boring thing they could do. I do feel though it's possible though that Joker never wanted Batman to die I think he wanted to share his immortality with Batman. I think he wanted a friend or partner to share his immortality with him escpecially since the concept of a friend was a recurring theme. However I do like the idea that Joker's own lie was his own undoing.
Like how Batman was calling him on his bluff and said if he was really immortal he had nothing to worry about or something like that. Honestly I also loved the Five different stories and if any and the stories told by the kids in the school. If anything I feel End Game gives readers the chance to decide Joker's past and that's why I like it. If I had to choose I'd say Joker is an immortal Dionysus(word got thrown around a lot which is fitting because Dionysus was the god of drunken style crazy partying and fittingly madness) human that's been around for centuries and became the basis for many myths. However as the Joker said once in the comics "If I have a past I prefer to be multiple choice." and I feel End Game Embodies this line which is what makes it a great story.