The Web Always Existed

You travel back in time to The Middle Ages or One Million BC, turn on your Time Machine's computer and... there's Google! Apparently the Web has always been there... humans just discovered how to access it in the early 1990s. This phenomenon results from filmmakers wanting to have their time-traveler show off the World Wide Boomstick without letting logic get in the way. However, anyone with a passing understanding of technology will realize that this implies the Web was always floating out there and that people just needed the right equipment to access it. (Which, we grant you, is a pretty nifty idea for a story gimmick, but this trope more often occurs because the writers just weren't paying attention.)

Another variation occurs if a time-traveler brings a TV or radio back in time and it plays current shows rather than the ones that existed back then. In fact, this trope refers to pretty much any instance where a technology that clearly relies on outside input to function is placed in a situation, usually through Time Travel, where that input could not possibly exist and it functions anyway.

This could technically happen in real life, assuming time travel is possible. However, it would require the machine's owner to not clear its history and all the files downloaded from the website in question. The computer obviously would not be able to get to anywhere on the Internet it had not stored. And of course it would require a source of electricity, if you go back far enough. Alternately, if communications between the two times are available, you could theoretically route the Internet through such a link - the computer in the past would call a computer in the present and ask for the page, and the computer in the present would deliver it back. The lag would probably suck, though.

But with the advent of really big hard drive spaces (A few terabytes would be sufficient to get a good portion of The Other Wiki, for starters), one could conceivably download big, giant portions of the internet that is relevant and bring it back through time for later access. Of course, you must be pretty darn Crazy Prepared if you actually do this in the first place.

(Of course, the Doctor could make a cell phone call through time, so theoretically...)

Examples of The Web Always Existed include:

Anime and Manga

Averted in Dragon Ball. Future Trunks remarks that the GPS in his watch is not functioning properly because the satellites it is supposed to rely on have not been built yet. However, he can make it work on an older type of satellite.
In the Mega Man NT Warrior manga, Lan and Megaman.EXE discover an ancient ruin on the internet that belongs to a civilization that existed upwards of thirty thousand years prior to the story, maintained by a being called Pharaohman; keep in mind that it takes place somewhere in 20XX.
Implied vaguely throughout Digimon.

Fan Works

One of the Common Mary Sue Traits.
- The protagonist of the My Little Pony: Friendship Is Magic fic Through the Eyes of Another Pony discovers that his smartphone still has internet access, completely inexplicably. To the author's credit, it's not just there solely for coolness; the Internet access is used to jump-start a plot peripety, and then disappears.

Film

Whoopi Goldberg's character in the TV film A Knight in Camelot could access the Web while in The Middle Ages. This doesn't even get into the lack of electricity...
Lampshaded in Enchanted. Shortly after Nancy goes to Andalasia, she gets a call on her cell phone. She says, "Rather good reception here," then throws it away.
- This might actually be justified if you take the view that the mobile phone signal can travel through the portal from Andalasia Castle to Time Square, giving her roughly the same reception at the castle as you'd get in Time Square.
  - Might even be further justified in that Andalasia is a "perfect" fairy tale world, where EVERYTHING good will end up with a happily ever after...even cell phone service.

Literature

Inverted in Dirk Gently's Holistic Detective Agency, where the telephone never works, except when the apartment containing it is in another place or time. Of course, the person at the other end is always in the "present" that the apartment left. Also, when the phone is finally repaired, the time machine stops working.
Seen literally in Macroscope, where the great information network kills anyone advanced enough to read it and primitive enough to still have violent tendencies, as a way of keeping the AOLers out of Usenet, as it were.
Somewhat averted in the Axis of Time series by John Birmingham. Computers on the task force of warships from Twenty Minutes Into the Future that arrives in 1942 is limited to whatever websites were stored in their local cache memories. Which, however, is pretty much anything the plot demands.
- Leading to a scene where Stalin berates his subordinates because they're citing "old Wikipedia articles" when he wants up-to-date information.
Inverted in the Harlan Ellison story "Jeffty is Five." The titular character has Psychic Powers that the viewpoint character at first thinks allows him to modify radios and television to pick up shows from the 30s and 40s. Then he notices that they are new episodes of those shows, that is, they are episodes that theoretically would have been produced had the shows survived to the present day. Jeffty also gets new issues of defunct magazines and comics in the mail with new stories written by long-dead authors.
Averted in the Michael Crighton novel Timeline. A character mentions that if they would have brought a TV or computer back to the times of the Hundred Years War, there would be no signal to receive and no electric current to power any of them.
In The Company Novels, the titular Company maintains operatives at most periods of human history and allows them considerable leeway and cheap comforts (albeit because their nominal administrators lack the subjective time or understanding to know most of what the operatives are doing). As a result, communication technologies such as radio and broadcast TV tend to be widely used throughout human history right up to the point where they get invented and regular people could start noticing them, and the broadcasters have eclectic tastes. No Internet (no need), but any outside time-travelers who brought a radio or TV along would likely be picking up modern programs for any definition of "modern".
Inverted in one Dave Barry column, where he talks about his car's radio being so old it played Winston Churchill speeches.

Live-Action TV

As said above, the Doctor in Doctor Who managed to make an ordinary Nokia 3200 cellphone capable of calling through time. Possibly justified by the TARDIS acting as a relay station. No word yet on what the roaming charges are...
- Or rather the long distance charges. Even mobile-to-mobile can't save you now, doctor! Muahahaha!
  - Free, of course, since the Cardiff Rift's use for 'pit stops' for the TARDIS is nothing less than free energy.
Parodied in The Young Ones when the house travels back in time to the Middle Ages and the TV is showing Jester Balowski's Medieval Torture Hour.

Jester Balowski: We'll be right back after this break. (He breaks the contestant's arm.)

Parodied on the Venezuelan sketch show Radio Rochela where a character travels to the XIX Century and tries to use her cellphone. The response? An automatic voice saying "We sorry, but the reception is unavailable until the next century."
Subverted on Supernatural with both cell phones and the Web.
Inverted on Eerie, Indiana—one of the weird things the main character collected over the course of that series was an old-fashioned radio that only played music from the 1940s.
The entire premise of the educational series Newscast From The Past, with each episode giving a dateline from sometime in the Middle Ages, and complete with "commercial breaks" advertising things that would have been in vogue at the time (although they still subscribed to the idea that Europe demanded spices to cover up the taste of bad meat).
In a remarkable pre-Internet example (at least what seems to be an example of this trope), in the Star Trek episode "City on the Edge of Forever", Spock, using what he even refers to as "stone knives and bearskins", is able to access newspaper clips and VIDEO (well, film footage) of two different histories (which are futures from 1930) with his tricorder!
- Actually, that is explained as he recorded the images the Guardian was showing before they went back in time. He just needed to use the tricorder to slow them down and isolate them. The real problem is that the few vacuum tubes that he bought in the 1930s would be of any help at all.
  - He had a Jacob's Ladder: the nifty V-shaped device that emits sparks. Known fact: Jacob's Ladders can do anything. As long as you have one on any kind of device, it will enable said device to do and access anything. Standard equipment in the Mad Scientist Laboratory.
  - Well, of course. A Jacob's Ladder is tamed lightning, after all...
According to Heroes, the internet has not only always existed, but is sentient.
The premise of the historical tv skit series History Bites is "What if Television had been around for the past 5000 years?", and gleefully mixed historical information with modern pop culture, such as Martha Stewart's advice on how to use pot-pourri to mask the smell of plague victims.
Averted in Journeyman, where the main character's cell phone refuses to work in the past. Instead, be finds his old phone from The Eighties.
Done deliberately on Fringe with the mysterious number stations. It is mentioned that when Marconi first activated his radio they were already transmitting.
Played straight in comedy sketch show Armstrong and Miller. A random guy goes back in time to meet a famous inventor and gives a spiel about the enlightened future, only when the scientist starts asking questions (such as "how do these radio waves work?") he realises he doesn't know the answer. Cue "What sort of a future is this?", so he pops back to the future, brings back a laptop and shows the inventor some porn. Answer: a good one.

Tabletop Games

Possibly the most spectacular example occurs in the Old World of Darkness setting's Mage: The Ascension game. A Technocrat faction of Magi called the Virtual Adepts discovered that the lowest levels of the Internet accidentally connected to a part of the spirit-world called the Digital Web, allowing skilled hackers to manipulate reality by sending data through the Internet.
- Note that within the setting, the virtual adepts manage this trope twice, as on top of discovering that the internet had its own spirit-world and so had always existed, they had in fact already invented a primitive text-only internet (roughly equivalent to UseNet) back in the 19th century. And as an aspect of the setting was that the internet as the mundanes experience it has always been a considerably backwards version compared to the version of the internet used by the VA, then if a mundane went back in time to World War I or World War II with the right equipment they might have full access to an equivalent of All The Tropes. Or their equipment would explode from paradox. Or it wouldn't work. We don't recommend you test it.

Video Games

In the Xenosaga series, the futuristic internet (the Unus Mundus Network) had always existed but could not be accessed until a network was layered onto it. The Unus Mundus ("one world")Network is some sort of hyperspace/collective unconscious/something network that's always existed, but they hadn't started using it for FTL web browsing until relatively recently. There's no implication that you could go back in time to see data from the future. It is also God and represents the theory of a universal collective unconsciousness.
Implicitly invoked in the Cimerora zone of City of Heroes. A set of islands reached by time travel back to the Roman era, Cimerora nevertheless has wireless coverage for hero and villain comm systems, including the ability to phone your contacts back in the 21st century. (Although this may be an unexpected side effect of the presence of other high-tech time travelers based within a nearby fortress. That still doesn't explain why the Roman in the street knows about the latest doings in 21st-century Rhode Island...)

Web Comics

The iPhone in The Way of the Metagamer works perfectly within the Medieval world of Dungeons & Dragons.
Inverted in Homestuck. The entire planet is destroyed by meteors, the homes of the SBURB/SGRUB players are transported into the game itself, and yet the Internet still works fine.

Western Animation

Hand Waved in We're Back! A Dinosaur's Story.
Inverted in The Simpsons; when Lisa sets up Grandpa's old radio in the living room and turns it on, it plays the Glenn Miller Orchestra and FDR's "infamy" speech.
- Something similar happened in a Crocodile Dundee film when the protagonist was shown a modern TV in a hotel room, he remarks that he remembers seeing a TV once a long time ago, turns it on, and... I Love Lucy comes on. "Yeah, that's about how I remember it".
Averted in the Futurama episode "Roswell That Ends Well," where upon being sent back in time to 1947, the Planet Express ship crashes because the satellite network that it relies on around Earth to navigate doesn't exist yet.
- Also in an inversion of this, their chronometer, that should work fine in the past, is turned into a pin-up calender.
Happens a few episodes of Time Squad with both internet and radio.
In the DuckTales (1987) episode where Bubba the Caveduck has his debut, Huey, Dewey and Louie introduce him to rock'n'roll. By bringing a radio into the stone age.