Buffy the Vampire Slayer has a whole bunch of stuff contained in non-show stuff. Hell, the universe is continued in the comics.
The miniseries Rose Red has a tie-in novel that fills in some of the background. Not required, but nice. Unfortunately, it refers to a website with lost excerpts from the novel (including one that implies Ho Yay between Ellen and Sukeena), but that site is gone.
Battlestar Galactica series finale infamously divided the fandom with its vague non-answers over the nature of 'God' and its refusal to explain either the 'Head-People' or what was going on with Starbuck. The Sci-Fi Channel had already approved a comic series called The Final Five which will address some of these questions: a preview of the first issue, for example, reveals that Pythia appears to have been an earlier incarnation of Starbuck and was given her visions of the future by Head-Six, whilst the father of Saul Tigh was one of the original Cylon skinjobs of Kobol. The comic is written by one of the TV scriptwriters with the approval of the producers: however, a caveat was added to the first issue stating that it was "An original interpretation of the story,", making its canonical status dubious. This may actually have been a wise move: having been denied their answers on TV, the dissatisfied section of the fanbase may have actually exploded to learn the real answers were being given in a comic book mere months later.
More overtly, there were two serialised webisodes (totaling 30 minutes each) preceding Seasons 3 and 4.5 which expand on many important elements. The Season 3 webisodes show life on Cylon-occupied New Caprica, how Duck and Jammer ended up where they were in the opening episodes of S3, how Tigh and Tyrol's morality was gradually eroded until they were willing to consider the use of suicide bombers and more. The Season 4.5 ones are even more important as they show exactly how Felix Gaeta lost faith in the battlestar's command crew, setting up later events in the series, and explain why he stabbed Baltar and lied on the stand in late Season 3. The existence of these webisodes is not well-known outside of the USA, as they tend not to be included on the Region 2 DVD releases.
The 4.5 webisodes haven't been included on any DVD releases.
It also confirmed everyone's suspicions about Gaeta's sexuality.
The true purpose of the DHARMA Initiative from LOST was only revealed in the Internet ARG The Lost Experience.
However, the writers have specifically stated this knowledge is not crucial to understanding the show; it's meant as a bonus for viewers who want more.
In addition, some of the material from the game has been given in the series in a different context, making tracking down information on the game not essential to understanding the story in the series.
The extra material involved with 24 is staggering. There are book series, one-shot comics, the prequels that are only included on the DVD release (and are actually important to understanding the context of the season), and information on the various forms of the 24 website at FOX.com (which gave background information on the characters never mentioned in the show) that are crucial to understanding some characters and their subplots:
Operation Nightfall, referenced by Jack Bauer and others during the first and third seasons, is only shown in the comic series Nightfall.
During Day 2, a feature on the official website references a report making the rounds that has sensitive information regarding Jack's experiences on Day 1. That report is the basis of the "24: The Special Subcommittee's Findings" book.
The 24 video game shows how Max, the mastermind behind the events of Day 2, is cornered and killed. It also focuses on Kim Bauer's first day on the job as a CTU intern.
The season four prequel shows Jack being fired from CTU, and with his new girlfriend, Audrey. The fifth-season prequel shows how Jack was discovered after faking his death for a year. While the Day 6 prequel is irrelevant, there is extra material included on the Season 6 DVD set that takes place after the day's events finish. Jack is debriefed by two fellow CTU officers, and gives background information on himself.
The show Babylon 5 left out the conclusions of some subplots, because they were going to be covered by the (canon) novels.
The Legions of Fire trilogy explained how we got from Londo becoming Emperor to Centauri Prime being ablaze 17 years later (as seen in various flash-forwards), whilst the Passing of the Techno-mages trilogy explained how Morden survived the nuclear explosion on Z'ha'dum and why the Shadows apparently didn't have any AA batteries protecting their capital city. Interestingly, whilst allowing the novels to expand and even resolve important plotlines from the show, JMS drew a line at explaining how the virus afflicting Earth in Crusade was eliminated in case he was able to revisit the series later on: a character in the Centauri books starts explaining it but gets interrupted.
In the first series of novels, both "The Shadow Within" and "To Dream in the City of Sorrows" are also considered canon. The first deals with what actually happened to the Icarus, Anna Sheridan, and Morden on Z'Ha'Dum. The other describes in more detail Sinclair's early days as the ambassador to Minbar, as well as the conclusion of Catherine Sakai's storyline, the first reactivation of the Rangers, and why Sinclair's hair is white in the whole Babylon 4 storyline. The second is especially tasty because it has a buttload of info, due to the fact that it's written by JMS's wife, Kathryn Drennan.
Also, the comics were considered canon. In a possibly ground-breaking moment, Garibaldi's out-of-the-blue announcement in the TV episode Messages from Earth that he saw a Shadow ship being excavated on Mars seven years earlier is actually a reference to a comic storyline produced a year earlier depicting his first meeting with Sinclair. Fascinatingly, this storyline also set up the departure of Talia Winters by revealing that the Psi Corps had been experimenting on her with Shadow tech at the same time. The first four issues of the comic also revealed all of what happened to Sinclair, and how he ended up as the Earth ambassador to Minbar. A miniseries called "In Valen's Name" described the final fate of Babylon 4 and showed some of Sinclair's tenure as Valen.
To understand some plot points in Heroes, you have to check out the online graphic novels.
Regarding the Doctor Who episode "Journey's End": The Doctor Who Adventures magazine reveals that the Doctor left Rose and his human counterpart with a TARDIS root(presumably so they can go off and have their own adventures) a detail which has a huge impact on the ending shown in the episode. The scene was shot and was boxed with the DVD set. Russell T Davies has stated that whether we acknowledge it is our own choice.
A much worse Doctor Who case is the notorious old school story "Ghost Light", which had almost all its exposition cut for time, making it almost incomprehensible without the lengthy DVD special features explaining the plot. Oh, and the DVD came out fifteen years after the TV broadcast...
From "The Angels Take Manhattan," There is no closure for Rory's father Brian; whether or not the Doctor ever goes back and tells him what happened is never said, despite him being introduced just three episodes earlier and thus being very fresh in viewers' memories. A scene was written, storyboarded and given a voiceover by Arthur Darvill, but not filmed, that gives him some closure .
Many of the new-gen Kamen Rider series are prone to this.
Although, this is usually minutiae such as weapon or attack names or height and weight. You don't need to know that Kamen Rider Kiva's Rider Kick is called the Darkness Moon Break and that it has the strength of 30 tons of TNT to enjoy the show.
Hyakujuu Sentai Gaoranger is subject to this. "Hyakujuu" is a Japanese word that means "All of the Animals". It literally means "One Hundred Animals". Seventeen show up regularly in the TV series, five make up the ultimate Humongous Mecha, another one debuts in The Movie, another shows up in a drama CD, and four more make their appearance in the Grand Finale. The other 72 show up in the series concept art, where a few of them were even conceived to be further Mecha Expansion Packs, become their own Humongous Mecha, and even more to have short one shot appearances in the finale but were left on the cutting room floor. These ranged from animals like the horse and cobra to the penguin and reindeer (named GaoRednose no less).
During the Disney era of Power Rangers, there was a lot of things that were placed on the shows website or in promotional materials that never actually showed up on screen. During Dino Thunder, Disney's site for the series said that Ethan developed a way for them to teleport across the city by text message. Jungle Fury gave Lily's name as Lily "Chill" Chilman, a nickname she was never called by. Finally, never once in the entire season was it mentioned exactly what RPM stood for (It's Racing Performance Machines, if anyone cares.) Whether it's All There in the Manual, Aborted Arc, or They Just Didn't Care is a matter of debate.
Farscape: Crichton's notes provide some musings and further information about various technology from the show. The Journey Logs, written from various characters' viewpoints, are also good sources of character insight, wit, and lampshade hanging ("Apparently Scorpy had been able to trace my DNA from the sample he took when he had me in that frelling Aurora Chair, and that enabled him to find my head. Don't ask me how that makes sense. I just work here.") in addition to being episode recaps.
Also, you have to play the video game to find out how Crichton came to have a favorite gun that he names Winona.
Twin Peaks had several print and audio media (The Secret Diary of Laura Palmer, Diane:The Tapes of Agent Cooper, e.g.) which offered teasing insight into the developing plot.
Firefly. While Joss Whedon probably would've explained it all had the show not been screwed over so badly, if you want to know Shepherd Book's past you need to read a comic book titled "The Sheperd's Tale", which was released in late 2010.
Although this history is hinted at pretty broadly in the movie Serenity.
The exact layout of the systems that make up Alliance territory was eventually revealed when the "Official Map of the Verse" was released. It shows that Alliance territory is divided into a complex, multi-starsystem cluster with five star systems and a number of smaller protostars, with four smaller systems orbiting one massive supergiant, around which the Central Planets orbit.
Degrassi: Whether the original or Next Generation, you always learn more about the characters from the website, DVD extras, and tie-in novels than you EVER would just watching the show. Melanie wanting to be a professional writer, or Liberty being a year younger than anyone else, for example...
The full rules of The Amazing Race are never fully disclosed during the show, and many rules are only brought up when a team breaks them.
A rather heartbreaking elimination by lost passport (rendered all the more heartbreaking by the fact that the team in question had finished the leg first before they discovered it was missing) was properly explained in an online video, where the team admits that they took a wrong turn and must have lost the passport in the dark (which also justified not showing it in the actual episode).
The motivations of the players of Survivor are often utterly inexplicable to those who have not watched the supplementary videos on CBS's website (and sometimes remain inexplicable even then until interviews with the players after the game has ended make things more clear). This is often the case for 24/7 Reality TV shows which only use a tiny percentage of their filmed footage to form the narrative of the show.
Some of Carnivāle's mythology was given in information and notes from series creator Daniel Knauf outside of the show, later presented by the fans via The Gospel of Knaufias. While it's not essential reading for the show, it does answer some mysteries and fill out some gaps concerning the show's plot and characters.
Played with on Warehouse13: Pete hasn't read the manual, Myka has read it cover to cover, and Artie doesn't need to.
Wizards of Waverly Place's website does more of an efficient job at explaining the functional magic rules and plots than the program itself does through dialogue.
Helix has supplemental material available at Syfy's Helix website, particularly the Access Granted subsection, which contains in-universe personnel files, video resumes, and cryptic documents.
The Mighty Boosh isn't a show heavy on continuity or deep backstories, but The Mighty Book of Boosh gives a few details that explain the characters a bit more. Among other things, it tells us how the Hitcher knew Jack the Ripper (Hitcher raised him as a sort of apprentice), and tells of when Feral Child Vince was first inspired to leave the jungle for London (A nature documentary crew filmed him being nearly murdered by a crocodile, which gave him his first taste of stardom).
Enforced by the PBS Red Book in situations where the sponsor count for a particular program (e.g. Eyes on the Prize and Independent Lens) is too high for the program to conceivably credit all the sponsors and square with the Red Book. The standard credit to use in such situations is "...and others (a complete list is available from PBS)."