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A rather unique example happens in the game Portal, where GLaDOS might have been modeled after a bound and gagged woman hanging upside down (as demonstrated in this fan art), making her even creepier.
Portal 2 revealed that this is very likely true: GLaDOS was formerly the secretary of the man who founded Aperture Science, whose dying wish was to have her brain uploaded into the computer mainframe so she could continue running the company in his stead. Some infamous Dummied Out dialogue suggests that this was done against her will.
Unusual for the Myst series, with its distinct lack of violence, in Myst IV, a preteen girl, Yeesha, is kidnapped and later seen, first in flashbacks and later in person, tied and strapped to a chair, with rope reinforcing the straps. In a flashback, she struggles mightily and appears to be crying. She's also played by a live action actress, due to the game's use of FMV crossed with prerendered backgrounds.
Happens at least once per game in the Monkey Island series...except for the first game. This is ironic because the first game actually centered around a kidnapping and subsequent rescue attempt.
iPod Touch game Wild Wild Train features cutscenes telling a story of a damsel in distress. The cutscenes are done using actual photographs of actual actors playing the damsel, villain and hero, all of whom are silent film stereotypes. And the damsel is tied up and gagged and blindfolded, quite well. It's enough to make you wonder... A collage of 9 of the cutscenes◊
Occurs near the end of Runaway: A Twist of Fate. Gina (one of the playable characters) ends up bound and gagged by the Big Bad; pretty standard, nothing really remarkable. However, something unusual happens when she escapes. Rather than taking off the gag completely, she takes it out of her mouth and wears it as a stylish neckerchief for the rest of the game. Fits in pretty nicely with the game's sense of humor.
After being knocked out by the Proto Badget, Miles Edgeworth wakes up locked in a small room and tied to a pole. While he's freed five minutes later by Kay, this scene spawned a lot of happy fangirls. This was after he was handcuffed when he's suspected of murder earlier that day. Poor guy.
In the sequel, when Yumihiko Ichiyanagi is kidnapped, he's bound hand and foot, gagged, and locked in a dark room at the back of a garage — and this after being crammed into a box and thrown into the trunk of a car. It looks a little goofy (embarrassing if nothing else), but is treated as genuinely traumatic.
In Red Dead Redemption, you can lasso and hogtie bounty targets so you can bring them in alive. Or you can do it to anyone, just for fun. You get an achievement for killing a tied-up woman by leaving her on the train tracks, old-fashioned villain style, as seen here.
In Rule of Rose, this happens to Jennifer in an early part of the game as well as in the opening cutscene. The scene repeats itself much later in the game, except that this time Jennifer is also gagged with a fistful of red crayons.
Imps: A bright red crayon just for you! Lots and lots and lots for you!
Has happened to Max Payne at least once in his career.
Max Payne: He had a baseball bat and I was tied to a chair. Pissing him off was the smart thing to do.
In Batman: Arkham City, you find Harley Quinn tied to a pole and gagged close to the end of the game. You can ungag/gag her as many times as you want. You have to ungag her once to unlock a side-mission. After that, it's just for your own amusement and to hear her yell at you to stop doing that.
Though not with the gag, Relius Clover during his Astral invoked this when used against some characters. Of course, this being him, it invoked a far worse image. The only person he ever gags was Valkenhayn and it was justified because restraining a werewolf without a gag is disastrous.
Lance, more "gagged" than "bound", at Vivi's salon in Brain Dead 13. And he will remain that way until you choose the facial.
in Super Dangan Ronpa 2, the fifth victim, Nagito Komaeda, is found tied up and gagged with a piece of tape. The most bizarre part of it? He did it to himself.
Nathan Drake, when captured by Rameses and his crew, finds himself tied to a chair at the beginning of chapter 12 in Uncharted 3: Drake's Deception. Using his fighting skills, he escapes. Sully gets tied up, as well, though it was actually a dummy dressed like Sully being used to lure Nate into a trap.
Princess Monica Grania Bilseia in Super Robot Wars Gaiden is captured by Saphine Grace, who ties her up and gags her with a ball gag.