R2-D2 and C-3PO enjoy monkeyshines between trilogies
When the 1985 ABC show Droids first came out, it was a light-hearted Saturday morning cartoon that featured Artoo and Threepio wandering around the galaxy. They moonlighted as cafe waiters and generally were involved in robotic pratfalls. What's pretty silly about Droids is the roundabout, overly meticulous way it was shoehorned into official Star Wars canon. Explains Wookieepedia:

In A New Hope, C-3PO says that Antilles was his "last master". However, in the Droids series, the droids have numerous masters before Captain Antilles. Star Wars: The Ultimate Visual Guide gives an official explanation for this continuity issue, mentioning that the droids were "accidentally separated" from Antilles, which is when the Droids cartoon happens, "before returning to Captain Antilles' ship, the Tantive IV".

Also, this animated series afforded them a crossover with Scooby fucking Doo:

C-3PO and R2-D2 appeared on a Saturday morning preview show in 1985 on ABC entitled The ABC Saturday Sneak Peek and Fun Fit Test. The special was designed to promote their new show Star Wars: Droids. The host of the special was Tony Danza. While they were on stage, the chest from The 13 Ghosts of Scooby-Doo showed up, allowing for a brief, if non-canon, Star Wars/Hanna-Barbera crossover.

I'm assuming Artoo picked up his cigarette habit (jazz or otherwise) after meeting Shaggy.

The FCC inadvertently turns the Ewoks into the Care Bears
Ewoks was the other big Star Wars cartoon of the 1980s. Like Droids, it aired on ABC Saturday Mornings. Unlike Droids, it survived for two seasons and evolved into cutesy derangement. Notes an Ewoks retrospective from Star Wars Insider:

In 1985, a climate of cuteness was suffocating Saturday morning television. One reason was the Reagan administration's relaxation of FCC controls on kid's shows, which basically made it possible for Saturday morning cartoons to turn into unabashed half-hour commercials for toys [...] This explains the laundry list of restrictions Ewoks/Droids had including: guns must not look like blasters, fires can only be started by magical creatures, physical contact must never include punching and hitting — just pushing and shoving, never strike a character on the head, and always have characters wear seatbelts in a landspeeder. Writers struggled to keep the show interesting. As [Ewoks scripter Paul] Dini put it, "TV animation goes through peaks and valleys, but at that point it had fallen into an abyss."

I'm assuming this explains why the theme song was changed from the first season's bluesy jam by Taj Mahal...

...into the second season's demon choir of alien teddy bears shrieking. This makes "Yub Yum" sound like the Benedictine monks.

Xenomorphs are running around the Star Wars Universe
In the 1996 Dark Horse Comics miniseries X-wing Rogue Squadron: The Phantom Affair, a xenomorph from the Alien series can be found moseying through the background. Speaking of which...

ET pops up in The Phantom Menace
Lucas promised Spielberg he'd toss them in after Star Wars paraphernalia popped up in E.T. Also, The Phantom Menace is approximately 45% more watchable auf Deutsch.

Darth Maul fights Luke Skywalker
Or at least a solid-state hologram of the bisected Sith did in a 2003 issue of Star War Tales. Also, a non-canonical comic book short story from 2005 had a totally not dead Darth Maul with bionic legs fight Obi Wan and Uncle Owen at the Tatooine homestead.

Boba Fett falls into the Sarlaac Pit three times
According to Wookieepedia, the bounty hunter has survived three separate tumbles into the Sarlacc Pitt. Expanded Universe scribes just keep tossing him in there! Once is understandable. Twice is poetic justice. Three times in a vaudeville routine. [Photo via Adam16Bit]

Artoo beeps for Jesus and meets Bon Jovi
We've all heard of the Star Wars Holiday Special — hell, I once held an entire party based on it — but did you know that the droids once made a 1980 novelty record celebrating Christ's birth? It was Jon Bon Jovi's musical debut! It ain't canon, but if it was, this album would've brought some dire theological implications to the birth of Darth Vader.

The entire Star Wars Universe participated in a Street Fighter 2 style tournament
Such was the plot of the abysmal 1997 Playstation game Star Wars: Masters of Teräs Käsi. Can a Tusken Raider's fisticuffs defeat Darth Vader's lightsaber? Yes. This is almost as bad as that recent game that lets you kill Chewbacca on Endor.

Darth Vader trades in his lightsaber for big honking medieval swords
Not to be outdone by Teräs Käsi, select editions of the 2008 swords-and-sorcery fighting game Soul Calibur IV included Darth Vader, Yoda, and Vader's apprentice as playable characters. In Vader's ending, he equips himself with two big flamboyant broadswords.

Luke's clone, Luuke Skywalker
Not to be confused with Luke Skyywalker, the lead singer of 2 Live Crew, Luuke Skywalker was Luke's evil mindless clone from various EU stories. I honestly prefer the names "Lew Skywalker" or "Leisure Suit Skywalker." Both are way more sinister. PS: I would also like to see an Extended Universe tale starring Luke Skyywalker (language NSFW).

Luke Skywalker, ghostly drug counselor
In the Star War: Legacy comic book series, Luke did the force apparition thingamabober to prevent his descendent to go cold turkey on death sticks:

Luke would later appear with his descendant and Cade's father, Kol Skywalker, in an attempt to persuade Cade to stop using death sticks and to warn Cade that there would be upcoming events that would envelop him and those he cared about, and that he had to be ready.

Familiar places to visit
In the Expanded Universe, you can visit Belarus, Seoul, and Camden. You can also order a Sonic Servodriver.

The Star Wars gang meets centaurs and werewolves
In the derided 1994 EU novel The Crystal Star, the Star Wars gang goes on a Star Trek-like adventure where they meet fantasy creatures and a being known as Waru, which can emanate "anti-force" (whatever the dickens that is). Notes Star Wars Blog's review:

For fans of Star Wars literature, the mere mention of the name stirs up opinions as diverse as "it's the worst Star Wars novel ever written" on one end of the spectrum to the more flattering "actually, it only sucks really bad" on the other. Written by Vonda N. McIntyre and published in 1994, few works of Star Wars literature have been so universally panned [...] McIntyre introduced many elements in her book that seem to have no comparable precedent in Star Wars, including centaurs and wyrwulfs (painfully pronounced "werewolves," I imagine) and, in my opinion, she does not spend enough time on these elements to make them readily intelligible or attractive. Two of the biggest culprits are:

Waru — a golden trans-dimensional being with anti-Force capabilities (take a moment to try understanding what that means or exactly entails);

Weird Luke — Luke acting completely unlike himself in no subtle way. This seemingly has to do with the "anti-Force" associated with Waru. Perhaps it also has to do with the nearby crystallizing star that gives the book its title, which is presumably acting on super-Force-sensitive Luke in a correspondingly antithetical way, though this conclusion never seems to be adequately suggested, either overtly (as in the narration or dialogue) or thematically, and Luke just seems like a plain weirdo.

Sentient velociraptors!
I mentioned this a while back in a piece on undying dinosaurs, but it's so weird it bears repeating:

In the 1994 extended universe novel The Truce at Bakura, Luke runs afoul of the Ssi-Ruuk, a reptilian species of totalitarian human haters who force the Rebels and the fractured Empire into an unlikely alliance. Mind you, this book takes place minutes after Darth Vader is burned in a funeral pyre. So yes, the split second after Return of the Jedi wrapped up, Luke and the gang fought fascist saurians.

Although they're neither technically dinosaurs nor futuristic, I had to include them. Here's a picture of Darth Vader dining with some sentient velociraptors. I pray to this image for power every day.

A huge thanks to Jonathan Wilkins and Wookieepedia. Top image via Slashfilm.