Star Trek IV: The Voyage Home is possibly the most beloved film in the original series cinematic universe. Directed by Leonard Nimoy, it blends humor, heart, and plenty of action as Admiral Kirk and his crew travel back in time to save the Earth from an alien probe (its motives weren't as clear as they are in the novelization based on the screenplay,) and possibly humanity itself.
While on Earth of the past, a few members of the crew did or said things that might have altered the past in small but significant ways, not for history at large but for a few 20th-century humans who crossed their path.
Chekov leaving his Starfleet ID and weapon with the FBI director
When Chekov and Uhura went aboard the naval vessel Enterprise to collect nuclear photons for their crippled Klingon Bird of Prey, the former was captured by the ship's security and taken to a room for interrogation by an FBI agent. During the questioning, he scoops up a phaser and threatens to stun the man. When the phaser fails, Chekov tosses it to the agent and escapes without recovering his ID, his communicator, or the phaser.
Imagine the FBI agent's frustration once Chekov vanishes from the hospital. Who was this man? Where did he come from? Was he just some lunatic LARPing, or was he some undercover agent sent by Russian spies? Did he chase Chekov's existence down the rabbit hole that he eventually became obsessed with solving the mystery? It's likely his supervisor gave him some "personal time" once the agent tried to spread the story; it may have even ended his career! (The phaser, communicator, and ID were sent to be studied by "top men", no doubt!)
McCoy curing the old woman's kidney failure with a pill
It's difficult to begrudge Dr. McCoy's actions when he encounters an old woman in the hospital while trying to help Kirk and Dr. Taylor prevent a possibly life-ending medical procedure doctors have planned for Chekov. McCoy is outraged by the mention of dialysis and slips the old lady a pill. About 15 minutes later, during the escape from the hospital, the old lady is sitting up and telling everyone that a doctor gave her a pill and she grew a new kidney that appears "fully functional."
While Bones might have saved and extended this woman's life, her survival might have changed the timeline in dozens of small ways. It's likely her doctor and even specialists examined and performed medical tests that exhausted her, or it's possible that all the money formerly spent on dialysis could go toward her grandchild's education, changing that child's world for the better. (I wonder how much longer that old woman lived? Did that pill give her a few extra years, or did she end up starring in her own personal version of The Green Mile?)
Spock neck-pinching the punk on the bus
While this is one of the funniest moments in The Voyage Home, it's also one where the aftermath could have spun off in a thousand different ways. For example, what if, upon awakening from the neck pinch, the punk thought Spock and Kirk were avenging angels who'd come to show him the error of his ways? "Repent, ye, and go earn your San Diego real estate license!"
Star Trek IV: The Voyage Home is one of the franchises' darlings, and for good reason. Dr. Taylor's arc is every Trek fan's dream, and who knows; the movie itself may have inspired young fans and made them actors, filmmakers, ecologists, and writers.