Attack on Titan: The creators of the anime were caught completely off guard by how popular it was, resulting in a three year gap before they were able to make a second season.
Kentaro Miura, the creator of Berserk, is practically legendary for this. When he is actually on schedule, new tankobon volumes of the manga come out roughly twice a year, not out of the ordinary considering the magazine it's published in, but he takes hiatuses a lot Ė most notoriously, he ended one hiatus to start another one just two months later. And Berserk is a chapter-a-month series. These breaks are likely necessary to stave off breakdowns.
D.Gray-Man has experienced several stops in production due to the author falling ill. After a hiatus in 2009, the series returned with a fifty page chapter... and another hiatus. However, from November on, it started running monthly rather than weekly. It went on hiatus again in 2012, and didn't start publishing again until July 2015, but now the chapters come out on a quarterly instead of a monthly basis.
Fairy Tail: Between chapters 396 and 397, with predictable reactions from the fandom.
Fist of the Blue Sky, the Prequel to Fist of the North Star can only enjoy sporadic and unannounced releases since 2005 due to Tetsuo Hara, its illustrator, becoming blind in one eye from a condition suffered by one in a million. Having lost depth perception, all his panels have to be re-drawn and inked by an assistant; the fact that his insisted style is realistic and intricately detailed does not help to alleviate the problem.
Hellsing: 10 years to finish a 10 volume manga. Kohta Hirano is (in)famously known for being quite a lazy mangaka when editors are not keeping constant watch on him. Near the end of the series it seemed Hirano would release a new chapter whenever he felt like it; he was often taking breaks from working in a monthly magazine. Many of the chapters in the second half of the series are just 9 to 12 pages long!
The OVAs also followed suit in what it seems to be a staple in the franchise, the first three episodes had just a few months between them (typical for a lavishly-animated OVA). From the 4th onward, the delays would just get longer and longer as Satelight was having a hard time producing the episodes all by themselves. After a year and half without releasing anything, Satelight passed the baton to Madhouse and it looked like they would keep up the pace, and they did... up to the 7th episode, another year and few months (and many rumors that Madhouse just dropped the series) is what it took for the 8th episode to get a release date; the series finally finished with the release of the 10th episode in late December 2012, nearly seven years after the series began.
At least the English release of the OVA's had an excuse (original licensor shut down during production of Episode 4; new licensor attempted to get the rest but a merger in Japan forced them to redo all the contracts from scratch, which took over two years).
The Toonami broadcast of the OVAs even managed to follow suit: due to a clerical error with the rights, the final 2 episodes weren't able to air until December of 2014, after the series had left off with the 8th episode in the second week of November.
Hirano topped himself with the prequel, Hellsing: The Dawn. After 6 chapters, he simply stopped in 2007. The Dawn didn't even have enough chapters to publish a single volume, so the series remains just a collection of 6 separate publications in 6 different Young King Ours+ issues.
Nausicaš of the Valley of the Wind was originally started in 1982 solely to get funding for the film (as people with money would not back an anime movie not based on a manga when Hayao Miyazaki was still a relative unknown). The final story was 7 volumes long (about 3½ years for a monthly series, and these volumes were shorter than most), but it did not finish until 1994. This is because Miyazaki would take long hiatuses to directsomemoremovies. The end result of this is that the story in the Nausicaä movie only covers the first volume and a half of the manga.
The manga of Neon Genesis Evangelion had been running since just before the anime started and it finally reached its conclusion in 2013, after 18 years of production. It averaged around one 6-stage volume a year (that's about one chapter every two months). There could occasionally be a full year in between chapter releases.
The Rebuild of Evangelion movies count as well. The entire series was supposed to have finished up back in 2009. The current target release date for the final installment? Winter 2015. Well, that time has come and gone, and now no one knows when it's coming out, thanks to Anno's involvement with Godzilla Resurgence.
The series of light novels of Haruhi Suzumiya have suffered of it as a whole: it had a hiatus of four years in between the releases of the ninth volume (April 2007) and the tenth and eleventh ones (May 2011). That, and the fact that the author once confirmed that the eleventh novel was not going to be the last volume...
In a similar vein, the second anime season got a release for summer 2009, when it was originally announced in 2007 for a Fall 2007/Spring 2008 release.
The Descendants of Darkness manga began in 1997 and updated fairly regularly until the release of volume 11 in 2003. Then there was a full seven years between volume 11 and volume 12- it wasn't released until 2010 (apparently due to problems with author/artist Yoko Matsushita's health)
Pretty Rhythm Aurora Dream was supposed to premiere on April 2nd, 2011, but due to the Tohoku earthquake and tsunami, the show premiered a week later.
Due to Too Soon issues related to the Tohoku earthquake and tsunami of 2011, the penultimate episode of Puella Magi Madoka Magica was postponed. Word of God expressed relief at the delay, as the earthquake cost the staff precious time that would otherwise have been used to make the episode. Eventually it was announced that the last episodes of the anime would air together about a month later (although this was apparently well after the episodes were completed). The last volume of the manga adaptation and the Prequel manga Oriko Magica were also delayed, the latter being moved from April to May.
JoJo's Bizarre Adventure had its 3rd arc licensed by Viz. It was scheduled to run over 16 volumes, which should've been a layup. Instead, the 11th volume was delayed from August 2008 to April 2009, for eight months, solely because of a scene where the villain, Dio, read pages from a book while threatening to kill Jojo. Why was this a problem? Well, in the manga, this book was just unreadable scribbles... but in the second OVA series based in this arc, writers copy-pasted text from the Qur'an in without knowing what it was, and the resulting protests caused Viz to halt all sales of all Jojo-related material, manga and anime alike, until they could be checked for other such errors.
Note that the actual manga series itself is from the We-Couldn't-Make-This-Up department but because of another reason. Hirohiko Araki, the author, has never missed a deadline in all 27 years of drawing Jojo (over 100 volumes and counting).
Mahou Sensei Negima!'s author, Ken Akamatsu, takes breaks lasting from a week to upwards of three weeks every now and then, claiming to be doing research. While he's almost as famous for his Shown Their Work moments as he is for inducing trope overdose, a lot of fans joke that his "research time" is spent researching his wife. In truth, as he has gotten older his drawing speed appears to have decreased and thus it takes slightly more than a week to draw one week's chapter, leading to a rough schedule of 3-4 chapters and then a break.
Akamatsu also took a lengthy hiatus about 3/4 of the way through Love Hina, so this isn't new territory for him.
Akamatsu apparently finally admitted an inability to keep any sort of weekly schedule anymore when he announced UQ Holder! would shift from a weekly magazine to a monthly one, mentioning his age as an involved factor online.
Newer fans of Guyver tend to think this is what happens with Yoshiki Takaya however he consistently puts out one book a year and has never once put out an issue each month for an entire year. He did once have a hiatus to finish up another of his works from before Guyver.
One reason for the Adaptation-Induced Plot Hole and somewhat abrupt ending of Fruits Basket's anime while the manga was still in progress? Natsuki Takaya broke her left arm. Her drawing arm.note The other reason for the abrupt ending was that, because the anime was produced while she was recuperating, Takaya was able to be very involved in the anime's production. This resulted in a lot of micromanagement and meddling, which thoroughly pissed off director Akitaro Daichi. Takaya didn't appreciate the pushback.
Then the third season didn't air until almost a year after the second, and there's still no word on a Season 4.
Digital Manga Publishing is infamous for this, as it can take years for them to release volumes. It's gotten to the point that many have assumed that they lost the licsenses when they actually haven't.
After an almost-three-year hiatus from June 2010 to March 2013, the Black Lagoon series has finally started releasing new chapters. But as of February 2014, there have been no new releases, except for May 2014 when Volume 10 of the manga was released... which not-so-coincidentally covers up to all the currently released chapters (87 as of this writing).
Hibiki no Mahou was published in a monthly magazine for twelve years... but due to Rei Izumi's health-issues, in this time it only released 29 chapters (34 if one includes the bonus-chapters). It took seven years between the release of volumes 2 and 3.