Yes, there are quite a few. Some explore a different side of the Joker where he has a moment of redemption and saves Batman. It often plays with the complex relationship between them and the idea that even the 'villain' can do a heroic act.
One great 'Batman Saves Robin' fanfiction is 'The Dark Knight's Rescue'. It really delves into Batman's determination and the emotional bond between the two. The author does an amazing job of showing Batman's skills as he battles through various villains to reach Robin. Another good one is 'Batman's Swift Save'. It's a shorter piece but has intense action scenes where Batman uses his gadgets to save Robin from a very dangerous situation.
One common theme is redemption. It shows that even a character as chaotic as the Joker can have a moment of good. Another theme is the blurred line between hero and villain. It makes readers question if the Joker is really a villain in that moment.
It could imply a story where Robin, Batman's sidekick, plays a crucial role in saving Batman within the context of fan - created fiction. This might involve Robin using his unique skills, like his acrobatics or detective skills he's learned from Batman, to rescue Batman from a dangerous situation.
Fanfiction.net is another great place. It has a large collection of fan - created content. Look for the Batman or DC Comics section and then search specifically for 'Batman Saves Robin' within that. There might be some real gems hidden in there.
Their relationship would become more complex. Batman might start to see the Joker in a different light, perhaps as a victim of his own madness rather than just a pure villain. The Joker, on the other hand, might be confused by Batman's act of saving him and could develop a strange respect for Batman, or he might try to use it to his advantage in some new, devious plan.
One scenario could be that Batman is poisoned by a new venom created by Scarecrow. Robin, using his knowledge of antidotes from his training, manages to find the right ingredients and save Batman. Another could be that Batman is trapped in a collapsing cave, and Robin uses his smaller size and agility to crawl through tight spaces and free Batman.
In this fanfiction, Superman might use his super strength to physically stop Batman from harming Robin. He could also use his heat vision to create barriers or distractions.
At first, Batman would be shocked and a bit confused. But quickly, his detective instincts would kick in. He would view the deaged Joker as a new and unexpected threat. He'd start gathering information about how the deaging occurred, in case it was part of a larger, more sinister plan. Batman would also be very careful not to underestimate the deaged Joker. He might try to isolate the Joker in a secure location to prevent him from causing chaos in Gotham. And he'd be constantly on the lookout for any signs of the Joker's old, devious self emerging from this new, younger form.