Regardless of which iteration you're talking about, Star Trek was always about boldly going where no one had gone before. That extended to a variety of worlds and universes beyond their own. The beauty of Star Trek is that it lends itself to any possible crossover by its very nature.
These intercompany events happened primarily in comic books and never on screen but they were still amazing. For any fan of the Star Trek movies and TV series, these eight comics and one book are more than worth following up on.
Star Trek/X-Men
Back in the 1996 comic book, the crew of The Original Series USS Enterprise under the command of Captain James T. Kirk came in contact with a ship virtually destroyed by an spacial anomaly. Spock scanned it and discovered near-human survivors. That's a polite way of saying they were mutants, namely members of the X-Men including Beast and Wolverine. Yes, that means there were two Dr. McCoys on the ship at the same time.
The comic also featured the Shi'ar, including Gladiator who punched the Enterprise. This led to Kirk uttering the line, "Did he just punch my ship?"
Star Trek: The Next Generation/Doctor Who: Assimilation2
Crossing Doctor Who over with Star Trek has an extremely natural feel to it. The two venerable science fiction franchises have have been around for several decades and have created seemingly endless amount of media. The Borg and Cybermen teamed up leading to an alliance between the Eleventh Doctor and Captain Jean-Luc Picard.
The Cybermen betray the Borg, causing them to join the alliance of the Doctor and the Enterprise. The interesting side effect of this is the Doctor recalling memories of this pairing is the Fourth incarnation teaming up with James T. Kirk.
Star Trek/Green Lantern: The Spectrum War
This story starts in an alternate DC Universe where Nekron is on the verge of destroying everything. Ganthet escapes to the Star Trek Kelvin Universe with one of each of the different rings, where his remains are discovered by the Enterprise. The rings find new homes among the Enterprise crew as well as other species around the galaxy.
It is soon discovered that many of the rings' previous owners have survived. They return, and recover their rings, using them to defeat Nekron in their new reality. Some of the new Lanterns stay with the Enterprise while the others escape into space.
Star Trek Vs. Transformers
The five-part series was visually interesting because it mixed the 1980's art style of the classic Transformers animated series and the 1970's art style of Star Trek: The Animated Series. Naturally, the Decepticons have shown up on Kirk's doorstep and are causing trouble. Fortunately for Kirk, Optimus Prime and the Autobots are around. Starfleet and the Autobots forge an alliance to defeat the Decepticons.
It's a pretty straightforward team up though at one point Starscream declared himself the new Klingon Emperor. That got weird. This series shouldn't have worked but IDW pulled it off.
Star Trek/Green Lantern: Stranger Worlds
The follow up to The Spectrum War, Stranger Worlds showed an alternate version of the Kelvin Universe where the various Lanterns had integrated into their new reality. Some returned to their villainous ways. Others, including Hal Jordan, joined Starfleet.
But none of them could imagine the threat they were about to face, specifically Khan with a Red Lantern ring. Though his exterior is often one of calm, Khan is fueled by rage and revenge. He is the most likely person in the Kelvin Universe to be offered a Red Lantern ring and the worst person in Kirk's life to have one.
Star Trek: The Next Generation/X-Men: Second Contact
The next time the X-Men encountered the Enterprise, it was the version under the command of Jean-Luc Picard. But this time the Enterprise ended up orbiting Earth in the 1990s, sending away team to Earth to find technology to repair their ship. Naturally, they encountered the X-Men who recognized members of Starfleet from their encounter with Kirk.
The Enterprise's arrival caused a rift in the space-time continuum, something Kang the Conqueror took advantage of. The heroes combined forces to fix paradoxes involving "Battle at Wolf 359" and "Days of Future Past."
Star Trek: The Next Generation/X-Men: Planet X
Unlike the rest of this list, Planet X is a novel though it is still a sequel to Second Contact. Captain Picard and the crew of the Enterprise were dispatched to a planet called Xhaldian to investigate an issue infecting the population. They were mutating into beings with impressive powers. Sound familiar?
Fortunately for Picard, the X-Men arrived to assist with this problem. Starfleet and the mutants partnered up yet again to help the Xhaldians and stop them from being used as weapons by another aggressive alien species.
Star Trek/Legion of Superheroes
Time travel is a familiar experience for both the crew of the USS Enterprise and the Legion of Superheroes. In this case, Captain Kirk led The Original Series crew to sort out changes to the timeline that neither team understood or knew the cause of.
Enemies of both groups stood in their way, including the Borg. That would be a new experience for Kirk since Picard is usually the one who has to contend with them. But even the threats were not what they were supposed to be once the timelines were messed up.
Star Trek/Planet of the Apes: The Primate Directive
This crossover was more than worth it just for the title. Once again, two classic science fiction franchises met up. Kirk and his Original Series crew discovered that the Klingos were trying to expand their empire into another reality. They teamed up with the apes who ruled Earth in that reality. George Taylor asked for help multiple times but Kirk refused, citing the Prime Directive.
But they did help remove the Klingons from their universe. Despite not buying the likeness rights, IDW still reached an agreement with Charlton Heston's estate to use his likeness when drawing George Taylor.
Star Trek: The Next Generation/Aliens: Acceptable Losses
Out of all the crossovers Star Trek has been involved in, this out seemed like the least likely to pan out. The version of space laid out in the Alien franchise could not be more different than what is seen in Star Trek. That might be why it was cancelled before it was published. This one actually could have been incredibly cool, depending on the story. Imagine what would have happened once the Borg assimilated the Xenomorphs. It was a crossover that had huge potential, and would have pushed Star Trek in a new direction. It couldn't be any more weird than Archie vs Predator.