There’s a crazy rumor in the book "A Clash for Kings" that Shireen Baratheon's father is Patchface, the family’s half-mad court jester who doesn't appear in the show. Cersei, Tyrion, and Littlefinger launch the rumor in retaliation for Stannis Baratheon sending letters all over the seven kingdoms claiming Joffrey, Myrcella, and Tommen to be bastards of incest.
Cersei originally wants to deny the rumors and take the tongue of anyone heard spreading them in "A Clash for Kings," but Tyrion advises her not to. Tyrion says it would look like the rumors are true. Instead, the rumor of Shireen’s parentage is agreed on and Littlefinger's whores along with Varys’ network of spies spread the rumor through brothels and pubs.
We’re still sad they killed Shireen off in the show since she’s still alive in the book.
By the way, it's a huge bummer they never showed Patchface. He's actually a pretty interesting character. Remember the song "It's Always Summer Under the Sea" that Shireen sings in season three? Well Patchface sang it in "A Clash of Kings," and it's a lot darker than it sounds. Patchface lost his wits after being drowned by a storm that killed 100 men, and his rhymes tend to be ominous and gloomy and some think they may be prophecies. Patchface is good friends with Shireen, though, so it's entirely possible she'd sing the song, too.