Introduction:For all you Star Wars fans out there, I have some shocking news! Palpatine could very well be Anakin Skywalker's father and Luke Skywalker's grandfather! Remember: all information used in this thread has been taken from official sources, but the conclusions drawn are all made by me, so don't blame George Lucas (for this thread, at least).
Everything I got here is from the second trilogy, but the first 3 movies are mentioned a little bit, though nothing from them has been used to explain why Palpatine could be Anakin's father. A lot of what I write here is based on assumptions that could be true or false, due to the limited information we have from the movies.
Now, to understand what I've written, you must have a pretty good knowledge of the Star Wars universe and all 6 films. In fact, don't even think about continuing reading if you haven't seen any of the films (including episode III) because this thread contains some big spoilers.
Episode I:The Trade Federation blockade Naboo. Qui-Gon and Obi-Wan, who have failed in their negotiations for Naboo, convince Queen Amidala to leave Naboo for Coruscant. Their ship, however, is forced to land at Tatooine, for a new hyperdrive. It is there that they meet Anakin Skywalker and his mother.
Shimi Skywalker told Qui-Gon that there never was a father for Anakin, that he was just conceived out of the blue. Qui-Gon himself wondered if Anakin was not created by the midichlorians themselves.
Meanwhile, the Jedi council is surprised that the Sith re-appeared suddenly and weren't extinct, as they thought they were. Since the Sith number only two, the Sith lord has to do some dastardly scheming in order to let the Sith power grow without the Jedi knowing. What exactly did he plan? That'll be told later.
The final senes of Episode I shows Obi-Wan defeat Darth Maul, thus gaining the rank of Jedi Knight and the right to train a padawan: Anakin Skywalker. The death of Darth Maul and the training of Anakin is VERY, VERY, important.
Episode II:The Clone Wars were very important for Palpatine, with Palpatine fighting for the Republic and Darth Sidious for the Separatists. The fact that Anakin and Padme falls in love and are then separated for long stretches of time during the Clone Wars is also quite important.
Episode III:Anakin has some pretty nasty dreams about Padme dying in child birth. Obviously, he wants to prevent this, making his attachment for Padme his greatest weakness, and an easy target for Palpatine.
Palpatine tells Anakin, in a theatre, that there was once a Sith lord, Darth Plagus, that could influence the midichlorians themselves and save the lives of his friends and loved ones. He manages to convince Anakin that the Jedi will be unable to pull off such Force tricks on Padme, and that only the Sith can teach one to influence the midichlorians like this.
General Grievous and Darth Vader share many things in common. Both are more machine than human, and both have been saved by 'modern medical technology'. Or have they...?
Anakin fights Obi-Wan and ends up losing the greater part of his body, as well as most of his skin. We don't know exactly for how long he stayed there near the lava, but when he was 'saved' by Palpatine, he was already in a pretty bad state. The birth of Luke and Leia, and Vader's operation, took place simultaneously (as far as I know from the movies). Padme, it is said, died of a broken heart. While this is very romantic indeed (mmm, that's right

), dying of grief is very difficult, if not biologically impossible. Yes, she was more than incredibly upset and sad, but maybe, just maybe, someone or something pushed her over the edge?
Episode IV:Luke Skywalker destroys the Death Star by shooting a pair of photon torpedoes in a way not even a computer could do. How? Ben Kenobi was with him in the X-Wing. How? The Force is a mysterious...er..force.
Episode V:Vader and the Emperor agree to convert Luke to the dark side of the Force. In his duel with Luke, Darth Vader tells him that he's his father. He wants Luke to join him and rule the galaxy together, father and son.
Episode VI:When Vader threw Palpatine down that big, deep shaft in the second Death Star, he was already seriously wounded from the Emperor's Force lightning. Luke was worried that taking of Vader's helmet would kill him, but Vader didn't mind. Did he really think he deserved to die, or was it something else that made him unafraid of death?
Main argument:I believe that Darth Plagus is Darth Sidious' old master. By killing his master, he, supposedly, gained Darth Plagus' powers, or he killed him thinking he could get his powers (they're the Sith, after all). With himself as the undisputed master of the Sith, Darth Sidious, who must've been a heck of a lot smarter than Plagus, realized that it would take quite some while to get the Sith back on top. He trained someone in the ways of the Sith, named him Darth Maul, and started using his newly found power over the midichlorians.
Sidious knew that the Sith needed to grow, but having more than 3 Sith at the same time would be too dangerous. The Jedi and Sith fighting and Force techniques are very similar and each could be adapted for the other. Why not let the Jedi train a young, inexperienced padawan for him, so that he could then get rid of whoever was his current apprentice, and take the Jedi as second Sith instead? Darth Sidious knew that Jedi start at a very young age because young children were 'purer' and 'better learners' than adults and teenagers.
Converting a padawan or a Jedi that started from a very early age would be too hard: they would not budge from their Jedi principles. And besides, Darth Sidious needed a capable person, one who was naturally talented with the Force, not some 'normal' kid picked up by some Jedi recruiter in the streets of Coruscant.
On which planet does the Republic and the Jedi council have little or no influence in? Tatooine. We don't know much of just how much Darth Plagus or Darth Sidious could do with the Force. But if they could turn back death, they can also create life. Darth Sidious influenced the midichlorians, with his knowledge of the Force, to create Anakin Skywalker and make Shimi his mother. Since he was created by the midichlorians themselves, Anakin (and all his children) would obviously have a very large potentional with the Force. Now, he just needed to be trained.
To be trained, he had to be picked up by some Jedi, at a late age. Therefore, Darth Sidious planned the Trade Federation's blockade, because he knew that Jedi would come to Naboo's aid. He instructed the nemoidians to kill off the Jedi (he secretly knew that couldn't possibly succeed) in order to force them onto the planet. He also knew that, once on Naboo, the Jedi would want to take Queen Amidala to Coruscant. He allowed this to happen, but not before planting a hyperdrive that would malfunction (he probably [and that is a big assumption, I'm afraid]) instructed all the droid fighters to go for the hyperdrive and then it would not surprise the ship's crew if the hyperdrive started to work badly). The safest planet for Amidala, Darth Sidious knew, was Tatooine.
Ben Kenobi was able to talk to Luke, despite being dead, and I think that Darth Sidious made Anakin believe that he would marry her the moment he put his eyes on her. He probably would've fallen in love with her either way (otherwise, it would be a pretty bad love story...I wouldn't trust in the Force when asking someone out

) but maybe Sidious made Anakin's love even stronger by telling it to him in his dreams or some kind of limited mind control.
Darth Sidious knew Obi-Wan's potentional strength with the Force, and that Darth Maul wasn't skilled enough to beat him, not if they were both using the dark side of the Force (Obi-Wan was obviously very angry, because of his inexperience, that Qui-Gon was killed). With his anger, which he could control because of his virtue, and his skill with the Force, Obi-Wan killed Darth Maul and was then able to take Anakin as his padawan and train him.
It's obvious that Darth Sidious continued to influence Anakin. It was Sidious that gave Anakin the dreams of his mother and Padme dying. Because he knew his mother was going to die, Anakin went to Tatooine. One event led to the next, and suddenly we have a lot of sand people dead. Episode II was Anakin's first step towards the dark side, if only for a moment. Sidious must have wanted to see how his future apprentice is shaping up. Why Sidious wanted to give bad dreams about Padme to Anakin is obvious. His fear for Padme's life is his weakest spot, and it enabled Sidious/Palpatine to convert him much more easily. Why Palpatine let Count Dooku die so easily in episode III should be pretty clear by now.
How in the world does General Grevious stay alive? His equipment may be advanced, but I think that it is Darth Sidious' knowledge of the Force and influence of the midichlorians that General Grevious is alive. Also, Palpatine probably killed off Padme, and blamed it on Anakin, using the Force. She wouldn't have died, but she was certainly very sick. Because of her 'broken heart', Palpatine was able to do to Padme what he could never do to a fully healthy person: take away her life with the Force. General Grevious' and Padme's examples both make it perfectly reasonable to assume that Anakin Skywalker was also hopelessly beyond the reach of medicine, and needed the Force. All this time, Darth Vader has stayed loyal to the Emperor, even though he knew he didn't kill Padme (despite being told he was the killer), that none of Padmes friends or the Jedi could have killed her, and that only Palpatine was evil and strong enough to do so. The Emperor held a vice-like grip around Vader's own life, and it was this fear for his own life that Vader never rebelled against him master.
Nevertheless, Anakin Skywalker is (probably) the Chosen One. There is still good in him, and when Luke (and maybe Leia) made that good come out, Vader stopped caring for his own life and decided to kill the Emperor. With Palpatine gone, so would the influence of the midichlorians Palpatine had. Darth Vader, without Palpatine, died of his original wounds in the duel on Mustafar. Why didn't Vader gain Palpatine's powers like with the case of Sidious and Plagus? Because Vader turned back to the light side, as a Jedi, and not a Sith.
Conclusion:I think this idea is a bit far-fetched, but all of the bits of the movies make sense if put together this way. There may be some bits of information from either books or other sources, but, as far as I know, nothing important has been omitted from the 6 movies.
This also means, however, that Sidious is not Anakin's biological father, but he just told the midichlorians to create him. He made the midichlorians create him the way he wanted, I think, but Anakin has no real father. Only his mother.