Even though the first story of A Sticky Situation remains solidly in the Digimon season three universe, the second story tosses several of the characters into Star Wars: The Clone Wars due to a dimension-traveling digimon called Parallelmon (a Canon digimon, not an OC of the author). It is very well written and very entertaining.
Bayonsiders has War from Darksiders and Bayonetta from Bayonetta meet up, have sex, beat up angels, beat up demons, and have sex while beating up angels and demons at the same time. All while trying to stay in continuity of both games.
Brother Bubbles. Space Marine Terminator finds himself in Rapture. Then meets a Little Sister.
A favourite for many seems to be Doctor Who, due to the remarkable ease with which the Doctor can be inserted in any story just by having the TARDIS land there. One of the most famous, The Man With No Name, is a Firefly / Doctor Who crossover, which some say should be officially added to Canon.
Yet it VERY strangely lists one of the unofficial rules of the internet dealing with crossovers (specifically, rule ⑨
Snakeskins is, without a doubt, one of the best fanfics in the whole crossover with Axis Powers Hetalia fandom, in which the two universes are blended amazingly and England and Italy are still their immortal Nation-tan selves while they're at Hogwarts. It... sound's kinda confusing, but it's a lot simpler in context.
Decent Pokemon/Digimon crossovers are not that common in fanfiction, but of late some interesting new submissions have popped up. Sturgeon's Law may be applicable due to grammar issues, but those with decent grammar tend to be alright.
The Ranma ˝ fandom loves to create crossovers. There exists several archives devoted to them. Particularly common are crossovers with Sailor Moon which has a sub genre called the Fuku Fic that has more fan fics written for it than most anime has fan fics total.
Shinji And Warhammer 40 K is sort of an odd example. It is not a story where the universes meet, being more of a For Want of a Nail story about Shinji getting a large amount of Warhammer 40,000 pieces and rulebooks as a child. His different (happier) childhood makes for a more dynamic protagonist and leads him to influence the rest of the cast into making things that fit the Warhammer 40,000 ethos more than the original Evangelion series did. It's not the straightest example of a universe to universe Cross Over.
A more straightfoward example of a mostly in-universe crossover for Evangelion/Warhammer would be Thousand Shinji.
This fanfiction crosses over Sonic the Hedgehog with the cast of My Little Pony: Friendship Is Magic, in a reimagining of the plot of Sonic Generations. It goes far beyond what the original game gives, showing the author did their research. For instance, Ponyville serves as the central hub, similar to the White Void in the game, the Death Egg Robot battle in the original is replaced with Nightmare Moon and in this fanfiction, Tails fights Metal Sonic instead of Sonic, possibly a reference to Sonic the Hedgehog 4. The author has even adapted the updated canon of Friendship is Magic into the fanfic. However, how the author intends to implement alicorn Twilight is currently unknown.
Star Wars and Star Trek. The rivalry between their fanbases and The Ultimate Showdown of Ultimate Destiny makes it a bygone conclusion feature some, and for those to require at least one major battle between the United Federation of Planets and the Galactic Empire, but there are certain exceptions.
Another good one is Unity by notable contributor SF Debris. What is a wormhole was connected from the Delta Quadrant circa Voyager's fifth season to the Star Wars Universe circa early Return of The Jedi, (disrupting the Rebel Alliance's attempt to attack the second Death Star.) Involved every character from TNG forward moving well into the future and exploring all the potential interactions in detail.
To complete the trifecta of renowned Star Wars vs. Star Trek fanfics, there's StarCrossed.
When With Strings Attached was first conceived, it was supposed to be a Beatles/Dungeons & Dragons crossover. Luckily, the author quickly matured out of that notion, and the story turned into something far more interesting and complex, though its RPG roots are quite evident still.
The longtime crossover champion for TV shows is The Witness by Michele Martin, which managed to cross Forever Knight, Highlander, The X-Files, Quantum Leap, and the Starman TV series. Believe it or not, the fic manages to keep all those balls in the air successfully, and even bring in cameos from a few other series, all while following a coherent plot and keeping the characters in-character!
In fact, Season 3 consists of entirely crossover episodes!
An author on F Fnet has created this  a Star Wars and Halo crossover where the characters of Republic Commando (and some friends and enemies) meet Master Chief and company in the Halo universe. There is a sequel that takes place mostly in the Star Wars universe during the events of the Original Trilogy.
In Anime fanfiction (and sometimes in other media), a quite common way to establish a plot, especially in Peggy Sue fics, is to have a wish be granted by Belldandy or her sisters. Whether this crossover is minor or major varies between fics.