You could focus on a character from one show ending up in the world of the other. Like if one of the Animaniacs gets lost and is found by the Doctor. It could lead to all sorts of interesting plot twists and character interactions.
You could write a story where she has adventures in time and space inspired by her love for the show.
You could write a story where the Doctor and the Muppets team up to save a magical world. It would be full of fun and adventure!
You could write a crossover story where the Doctor teams up with the Avengers to save the universe from a combined threat.
You could imagine a scenario where Doctor Who and The Avengers team up to save the universe from a time-traveling threat. Maybe there's a villain manipulating timelines, and both teams need to combine their skills and knowledge to stop them.
You could have a fanfic where they team up to face a common enemy in a new adventure.
You can find all sorts of stories. Some might focus on alternate universes or new adventures for the characters. Others could explore deeper relationships between them.
One possibility is a tale where a Time Lord gets involved in the Game of Thrones' political intrigues and uses time travel to change the course of events. Or imagine if a White Walker was accidentally transported to the Doctor Who universe and caused chaos.
You could write a time-travel adventure featuring Alexander Holmes and the Doctor facing off against a powerful alien threat.
You can expect to find fan fiction that imagines new challenges and mysteries for them to solve. Also, there are those that delve into their emotional bonds and personal growth. Sometimes, alternate universes or time travel scenarios are explored too.
You could write a story where they go on a new adventure in a previously unexplored time and place.