In Doctor Who, are there any instances of anyone "accidentally" travelling through time, i.e. not using a Tardis, vortex manipulator or any constructed time vehicle? I don't count people (such as Captain Jack) being able to travel through the Time Vortex by clinging to the outside of the Tardis.
In Earthshock, the sixth story of season 19, the occupants of a freighter in warp drive heading straight for Earth attempt to override the controls to take it out of warp. They successfully manage this and take the ship out of warp speed but accidentally set the ship into a time warp and go back in time.
BERGER: There's an override on the navigational control. My instructions are instantly countermanded.
BRIGGS: It's this thing that's causing it.
SCOTT: Can't you disconnect it?
BRIGGS: It's probably booby-trapped.
ADRIC: Maybe so, but it can be disconnected.
ADRIC: That's it! The first one's solved.
BRIGGS: We're running out of time. The troopers can't hold the Cybermen much longer.
BERGER: We're working as hard as we can. It's very complex.
ADRIC: It worked! See if it's released anything.
(She hits the final button and the freighter tilts. The Cybermen fall over.)
(On the scanner, the image of the freighter flickers then vanishes.)
NYSSA: What's happening?
TEGAN: Where have they gone?
DOCTOR: The freighter's jumped time warps.
BERGER: We're travelling backwards in time.
BRIGGS: That's not possible!
Well, honorable mention to the episode blink.
Sally and Larry return to the house with several of the DVDs and a portable DVD player. As Larry starts the playback of one of the DVDs, Sally discovers she can converse with the Doctor in the past, as he possesses a copy of the complete transcript that is currently being compiled by Larry as he listens. The Doctor explains that he and Martha were transported to 1969 by the Weeping Angels.
Still am very suspicious when I see an angel statue hiding its face.
Then there are time fissures, possibly very dangerous, possibly very fun if you're the doctor and like to shout geronimo
The Eleventh Doctor falls through the fissure and lands in front of the Tenth in the sixteenth century. Stunned, the Tenth Doctor dons the fez himself. The Eleventh pops up and gabbles excitedly about how skinny his predecessor is, which makes the Tenth realise who he is.
Then there is the lovely story of the girl in the fireplace
The Doctor, Rose, and Mickey explore the ship and are puzzled to find an 18th century French fireplace. When he looks through the fireplace, the Doctor sees a young girl and asks who she is. She replies that her name is Reinette and that she lives in Paris in 1727. The Doctor deduces that the fireplace is a time window, a device that allows direct access to another time and place.
The tragic story of The Girl Who Waited
After this conversation, the older Amy informs Rory that she is going to "pull time apart" for him, but that she will only help if the Doctor agrees to let her travel in the TARDIS alongside her past self. The Doctor reluctantly admits that the TARDIS could sustain the paradox. Through a hijacking of the regulator valves and by insisting that both Amys concentrate on a powerful memory — which turns out to be their first kiss with Rory while dancing the Macarena — the Doctor is able to pull the younger Amy into her future self's point in the time stream.
The Third Doctor story Day of the Daleks features the Doctor accidentally pulling a future version of himself and his assistant Jo Grant a short time into the present. At this point in time, the Doctor's knowledge of time travel had been blocked and the TARDIS disabled, so the time travel was accidental, although caused by the Doctor experimenting with the TARDIS console. The story also features personal, hand-held time travel devices called time transmitters used by soldiers 200 years from the future, but this time travel is deliberate.
The Fourth Doctor story City of Death featured an alien, Scaroth, who had been "splintered" into 12 pieces through time by an explosion millions of years ago, with one version living around 1505 and the latest around 1979. Other splinters are shown to be living in the Neanderthal days, ancient Egyptian times, and classical Roman times.
The Fifth Doctor story The Awakening featured a boy, Will Chandler, from 1643 who was pulled through time into 1984 by a temporal rift created by The Malus.
The Sixth Doctor story Timelash featured the Timelash, a device for disposing of political enemies of the state. It was basically a one-way time corridor. The creator of the Timelash either didn't know or didn't care what happened to someone thrown into it, however, they were not killed but ended up in Scotland in 1885. Vena, the daughter of the community's leader, is thrown down it, but is returned to her planet by the Doctor travelling the wrong way through the time corridor in the TARDIS.
The Seventh Doctor story Silver Nemesis features Lady Peinforte who, with her servant Richard Maynarde, travels through time from 1638 to 1988 by drinking a potion. However, this was intentional time travel, just not by conventional means.
How about the Confessional Dial from S09E11 Heaven Sent.
It is not intended as a time travel device but the doctor successfully uses it to travel billions of years into the future.
The doctor is attempting to escape a trap without revealing secrets. The solution he uses essentially puts him in stasis. The last iteration is aware that many others have been before him but has only experienced the time it took him to solve the puzzle the last time. He accidentally travelled billions of years forward in time.
Ace, a companion of the 7th Doctor, was accidentally transported through time by a 'time storm' as mentioned in the episode Dragonfire:
"Ace, a young woman who turns out to have actually come from 20th-century Earth, propelled forward in time when a mysterious time storm appeared in her bedroom while she was trying to experiment with "Nitro-9", an explosive of her own creation."
(Shout-out to Jules who also mentioned this one in a comment).
There are any number of occasions where a person has time travelled without his0 her express permission. There are two that immediately come to mind (both from the 17th century). The first person is Will Chandler in The Awakening who comes from 1643.
The second person (although whether he travelled against his will is a matter for some debate) is Richard from 1638 and was the servant of Lady Peinforte who travelled to 1988 via a potion that she devised in Silver Nemesis.
Yep here is a instance of "accidental time travel" in Invasion of the Dinosaurs.
The Doctor and Sarah arrive in 1970s London to find that the city has been evacuated, due to the mysterious appearance of dinosaurs. The monsters are being brought to the present by a disaffected scientist, using a time machine nicknamed the Timescoop, as part of a plan to revert London to a utopian pre-technological age. The plan calls for Central London to be depopulated prior to being moved back through time, so that only an elite group (who will re-found the human race in the remote past, along more ethical lines) will make the trip. This plan, known as Operation Golden Age, is secretly being supported by key personnel within the British Government - and even within the United Nations Intelligence Taskforce, which is ostensibly combatting the menace.