In real life, we're all the heroes of the stories that are our own lives; or, depending on the health of our self-esteem, the anti-heroes. When you look at it that way, any human interaction at all is a crossover. I think this is why I've always been fascinated by crossovers: the meeting of fictional "universes" is symbolic of the meeting of perceptive "universes" that happens to and all around us almost every moment. And DOCTOR WHO's time machine the TARDIS, with its total mobility throughout space and time, is the ultimate dramatic device to facilitate crossovers.
My self-esteem (like me) is healthy if even a bit overstuffed. In this regard at least the Doctor and I are very like each other. Now, another staple of fanfiction is the "Mary Sue" character, stereotypically a character modeled on the fanfiction author (stereotypically female) who is put in with the author's favorite property's heroes, with some sort of superpowers (e.g., in STAR TREK fanfiction, telepathy), saving the day for the series heroes (over and over again if appearing in a series of stories), and falling in love with and marrying the author's favorite character. When I was an adolescent STAR TREK fanfiction writer in the 70s, I created such author insertion characters - but not since I started watching DOCTOR WHO. With the universal exception of the final action attributed above to the stereotypical Mary Sue, this is also the description of every legitimate DOCTOR WHO story ever produced.
Those who don't understand why people write fanfiction instead of creating their own original characters will undoubtedly think it doubly sad that I should have found even my perfect Mary Sue already created for me. Interestingly, the only crossover in my fanfiction index that doesn't include the Doctor (though he is a significant offstage character) is between two of my wife's favorite shows, FOREVER KNIGHT and BUFFY THE VAMPIRE SLAYER.
The overall theme for my series of DOCTOR WHO/STAR TREK crossovers (though it isn't tapped for every story) is the conflict between the Starfleet captains' non-intervention Prime Directive and the Doctor's more proactive philosophy and actions. It's a theme that near as I can tell will never run dry nor ever be resolved, because near as I can tell they're both right - at least for themselves. And here we come back to crossovers as real life, because in every real life conflict, at least at the beginning, everyone involved believes s/he's the one who's right.