Naota's older brother Tasuku, only appearing as a silhouette in an episode 2 flashback and as a picture with his American girlfriend in a later episode, which is of course obscured by glare.
Medical Mechanica remains a near-complete mystery throughout the series. We never see anything of them other than the giant clothing iron in the center of the city, and the occasional killer robot that pops out of Naota's head.
The Pirate King Atomsk nearly becomes this trope, only being alluded to with a mysterious symbol on Canti's monitor and Naota's forehead whenever he's harnessing his power, with the first time he's even referred to by name being at the end of the second-to-last episode. He makes his first and only appearance in the last few minutes of the season. In FLCL Progressive he appears during the end credits of each episode, but viewers who haven't seen the first season might not recognize him since he's back to being referred to in vague terms and hasn't been mentioned by name.
Mr. Ichinose started out like this, but he did eventually appear (in a story where his appearance was the main focus). Oddly enough, he recognized everyone from Maison Ikkoku on sight, and knew many things about them. However, none of the tenants recognized him. He continued to make appearances in expected places for the rest of the manga's run.
Kyoko's deceased husband, Soichiro. While partly a Posthumous Character, even when he IS shown in flashback any exposed skin (face, hands) is blacked out. In universe, Godai, Kyoko's new love interest, never even gets to see a picture of him until nearly the end of the series when he sees a picture from Kyoko's first wedding. Even then the READER only gets to see a part of Soichiro's hand.
The sensei who Shinji lived with before the series began; we don't even know their gender.
Lina's older sister Luna in Slayers. She's mentioned several times (especially in TRY), but we never do see her on screen, or in the novels that the anime was based on. Word of God says that she'll never appear in canon outside of the opening sequences, because she's so ridiculously powerful that she would unbalance the rest of the cast.
Himeko from SKET Dance will occasionally make a passing reference to "Nakatai-san from Class A". Late in the series, Bossun finally points out that he has no idea who that person is. This leads to an entire chapter where he tries to meet the character, only to keep missing her due to increasingly improbable events. He does manage to catch a partial glimpse of her at one point.
Contrary to popular belief, Yugi Mutou from the original Yu-Gi-Oh!did have a mother, and she wasn't dead. She was seen in one brief scene in the manga, and one scene in one episode after the Duelist Kingdom arc (which was cut from the dub) but that was it. Exactly why she kept herself hidden all the time wasn't clear.
Jonouchi's father was someone who was never seen, except in one panel in the manga, but he was talked about often. Apparently, he was an alcoholic whose illness had caused his marriage to end in divorce. Jonouchi's mother got a small amount of screen time; she kept custody of his sister Shizuka, and was very reluctant to talk to Jonouchi until Shizuka's surgery.
Leina's parents, Owen and The Head Archangel are only mentioned in all the adaptations of the Queen's Blade franchise, except in the animated ones when they physically appear for first time, albeit Leina's mother previously appeared in a non-canonical light novel side-story.
A Certain Magical Index has the #6 Level 5 Esper. Known only by process of elimination for most of the story, it's eventually mentioned that they do not appear in public, meaning their gender, appearance and presumably even ability are not known by anyone. Thirty volumes into the story, their name was finally revealed to be the Gender-Blender Name Aihana Etsu, but even that wasn't confirmed to be their real name for another half a dozen volumes. Multiple people have impersonated them, a process that Etsu actually encourages for unknown reasons.
Mahiro's father in Haiyore! Nyarko-san; when his mother Yoriko first shows up, she implies that Dad'll be stuck at the office for a while making up the time they took off for their most recent honeymoon. As of the end of the series, his only appearance is in the second anime season during a Flash Back to his and Yoriko's courtship, and even then his face is obscured.
In the first episode of Pokémon, Ash winds up with Pikachu because three other trainers beat him to Professor Oak's lab. One was Gary, the other two are never seen or even given names. However, through the start of season one trainers and gym leaders Ash meets tell him he doesn't measure up to the three trainers from Pallet Town ahead of him. Oak eventually tells Ash and Gary that the other two weren't successful in getting the required badges.
The second episode is the only time Ash's dad is mentioned as Mrs. Ketchum says he would have been proud of him.
Karel and Liera Harlaown from Magical Girl Lyrical Nanoha Strikers are mentioned several times, but the only time the actually appear is in a family photo that their aunt Fate keeps by her bedside.
Kaguya-sama: Love Is War has the General Affairs and Auditing officers in the student council. According to the volume 4 extras, they're both third year students that aren't on good terms with the rest of the council, so they do their work separately. Following Shirogane's reelection in volume 7, the Auditing officer was replaced by Miko and the General Affairs position is presumably unfilled.
The father of protagonist Izuku Midoriya, of My Hero Academia, exists, is working overseas and can breath fire. That's about all we know about him.
Aside from a single note left for Yomiko, Nenene is completely absent from the Read or Die OVA despite playing a major role in every other adaptation.
The Sunflower Samurai, Fuu's father, of Samurai Champloo is obscured and we don't get a good look at him until the very end of the series. Information is very scarce with only subtle hints about his nature and whereabouts until the final third of the series.