WMG / Dragon Ball GT

Due to a quirk with Saiyan/Human half-breeds combined with a one-in-a-billion chance, Pan is Half-Saiyan.
As You Know, you get half of your DNA from your mother and half from your father. So, there is a small, minute chance that the lucky sperm (which has half of the father's DNA, just like the egg has half of the mother's) had only the DNA passed down by Goku to Gohan. However, this would still prevent her from being half-Saiyan, due to the fact that an X would have to come from both parents. However, we do not know much about Saiyan DNA, or how it interacts with human DNA. We do know that a Saiyan immersed in Saiyan DNA heals faster, which means it doesn't follow human biology. So, perhaps due to a quirk with Sayian DNA, the sperm has both gender chromosomes (having two copies of the same X for females), and actually modifies the egg to remove the one that's there. Perhaps both her gender chromosomes are Gohan's X, which, following this weird quirk of biology (or, if you perfer, Ass Pull of mine) would also be Goku's X, which means she would be Half-Saiyan (and genetically, equally related to Goku and Gohan). Now, if her child (the mother or father of Goku Jr.) was fathered by Trunks (who likely died with seemingly every other Z Fighter), that kid would be half-Saiyan, so their child would be a quarter Saiyan, allowing Goku Jr. to go Super Saiyan. As for why Vegeta Jr.'s mother doesn't know Pan, perhaps the families (well, either Bra and Bulma or likely just Bra when it comes to the Briefs') had a falling out over the fact that she was the only fighter to not die, including Vegeta and Trunks.

Alternatively, something bad happened about ten years before the finale that caused a falling out between the Briefs family and the Son family.
The fact that Pan is raising her grandson suggests something happened to Goku Jr.'s parents. This could be related to the reason why Vegeta Jr.'s mother (presumably Bra's daughter/granddaughter) doesn't recognize Pan. Something terrible happened that killed Goku Jr.'s parents, and the two families blamed each other for it and stopped speaking to each other.

Eis Shenron attempted to rape Pan.
Eis Shenron was extremely powerful, much more powerful than Pan. Plus, Pan was already weakened. So, how did he not easily kill her? Well, notice her clothing? It's ripped up alot more than it should be, seeing as Eis's powers can basically be summed up as "freeze and hit". Not to mention her pants/shorts are the focus of most of the damage. Finally, as revealed outside the anime, Eis is the embodiment of the wish to remove the memories of Majin Buu. So, he's the living embodiment of about 6 billion mind rapes. So, physical rape seems pretty much within his character, seeing as that's the physical equivalent of mind rape.
  • This makes sense. Squick.
    • He calls it his "reverse popsicle".
Pan was an accident.
Pan was 5 (or 4) about 5 years after the Buu Saga. This means that Videl had Pan while she herself was still a teen, perhaps still in high school. Something tells me that wasn't on purpose.
  • The Jump between Buu's defeat and Goku meeting Uub was 10 years. Pan was born 5 or 6 years after Buu died and was 4 or 5 when the tournament was held.
    • According to Battle of the Gods, you're right. Gohan didn't even know Videl was pregnant with her for a while and this was before they were married.
Pan's strength is just a bit abnormal for someone who's life has been pretty safe, even though she is part Saiyan. However, her father underwent a magical power-up, and then, about a decade later, another magical power-up (the Guru potential unlocking, and whatever Old Kai did). So, perhaps she got half of both of those. That would explain why she's stronger than Android 20, who in turn was stronger than Frieza (man, Frieza was pretty pathetic in hindsight, seeing as he's weaker than a 10 year old who hasn't even gone Super Saiyan).
  • While we're at it, how many times has Pan survived things that should have been fatal to someone at her power level? Well so did Hercule! Who says it's just saiyans that function on Lamarckian inheritance?
The entire series was all a dream.
Because no WMG page is complete without this theory.
Emperor Pilaf is a Time Lord
Same reason as the above, and hey. It would explain his presence.

GT is set in a reality where they didn't meet Beerus, and Frieza didn't get revived.
It'd explain the discrepancy pretty well, and actually makes the series make more sense (such as by illustrating that Pan has the potential to go super saiyan (see BoG), but simply never triggered the transformation in the GT 'verse, which fits a lot better than people thinking she just can't transform at all).

the shadow dragons are supposed to grow in power over time
At the point Goku and Pan hunted down the shadow dragons, the dragons were ridiculously weak by the standards of the current series, menacing only one village or city compared to as early as king piccolo or the red ribbon army which terrorized nations and nearly all the Z big bads who could destroy entire planets. The shadow dragons were meant to grow over time, training themselves like goku and honing their powers until they could fight stronger opponents and reach stages where they become Physical Gods like the eternal dragons to destroy planets by being near them and eventually travel to other universes

The GT universe has a much lower power scaling
building on the above, everybody in this series is generally weaker compared to the extent of previous series. For example: Baby tried to destroy earth personally during his fight with goku. At this point in the fight Baby had not only possessed Vegeta, but also turned into a super saiyan oozaru form which has an intense multiplication in Vegeta's power. His method of destroying earth was hitting it with a powered up form of Gallick gun, repeatedly, on a planet which had already started to violently break down thanks to the black star dragon balls. Maybe the writing stuff downscale the power so they could have more impressive fight scenes
http://tvtropes.org/pmwiki/pmwiki.php/WMG/DragonBallGT