Another storyline could be that Robin is adopted by another superhero or a group. For instance, he could be taken in by the Titans early on. This would change his character development as he'd be influenced by different people and their ideologies. He might develop a more team - based approach rather than the mentorship model that would have occurred with Batman.
Ra's al Ghul could also be a main villain. He might see the absence of Robin as a sign of weakness in Batman's legacy. He could try to take advantage of this and further his own plans for Gotham, perhaps even trying to recruit Robin for his own purposes if Robin is out on his own.
Well, in some fanfics, Batman might not take Robin for various reasons. One could be that the writer wants to explore a different dynamic in Gotham without the traditional Batman - Robin partnership. Maybe they want to focus on Batman's solo adventures more, showing his loner side and how he copes with the city's crime all by himself.
Maybe in that fanfic, the writer wanted to explore a different dynamic. Batman might be depicted as being too jaded or having some dark secret that prevents him from taking in Robin.
One implication is that Batman's character could be seen as more isolated. Without Robin, he doesn't have that youthful, optimistic presence by his side. This could lead to a more brooding and self - contained Batman. For instance, in some fanfics, he might become even more reclusive, having no one to share his thoughts or strategies with in the way he would with Robin.
One possible alternate storyline could be that if he had come, he might have joined forces with Batman earlier to take on a new and more powerful villain. This could have led to a different dynamic in the Bat - family, perhaps changing the way Robin was trained or how Gotham was protected.
One interesting storyline could be Robin growing up and struggling to step out of Batman's shadow. He wants to prove himself as an independent hero while still respecting Batman's teachings.
One interesting storyline could be baby Robin getting lost in Gotham and Batman having to search every nook and cranny to find him. It could show Batman's desperate side as he's not used to being so worried about a young Robin who doesn't have his full skills yet.
In some fanfics, Batman might attack Robin due to mind control. For example, if a villain like the Joker manages to get his hands on some new technology or a powerful magic item that allows him to manipulate Batman's mind. Another reason could be a misunderstanding. Maybe Robin was framed for something he didn't do, and Batman, in his haste to seek justice, attacks without fully understanding the situation. It could also be a part of a darker, alternate universe story where the characters' personalities are twisted, and Batman has turned to the dark side for some reason.
Perhaps Goku would form a different kind of bond with his friends. Instead of a marital relationship with Chichi, he might develop a closer brother - like relationship with Krillin. They could go on adventures together, not having to worry about family responsibilities. Goku could also focus more on mentoring younger fighters, like Gohan, in a more detached way, as a master rather than a father.
A really cool storyline could be a time - travel one in the fanfiction. Batman, Robin, and Batgirl are sent to the past or future Gotham. In this new time period, they have to adapt to different versions of the city and its threats. Robin might be especially fascinated by the differences in technology and crime - fighting methods, and together they work to find a way back to their own time.