PredictionBook registry - take one prediction a day to keep the hindsight bias away! - based on the speculation:

Harry's solution will be...

• nothing
• invoking Lucius's debt to whomever killed his supposed Imperiuser
• invoke the Imperius-debts of other Wizengamot members
• make Dumbledore turn himself in to Lucius in exchange for Hermione's freedom
• threaten to destroy Azkaban
• reveal the truth about Bellatrix's escape
• offer the Wizengamot Voldemort's true identity (Quirrel)
• accept Hermione's imprisonment but order the Dementors to leave her alone
• destroying one of the Dementors

• using the Time-turner

  • to alter Hermione's testimony to something easily falsifiable
  • to affect Lucius in some way
  • to smuggle Hermione out under the Cloak
• ask for a trial by combat
• have Quirrel make Bella confess to the crime
• confess to the attack
• demonstrate Hermione's wand did not cast the curse with Priori Incantatem

(These are not all mutually exclusive, and I didn't set down and make them all sum to 100%.)

Um. It's not like Dementors come with cloaks, you know. If cloaks-that-have-been-worn-by-Dementors were really valuable potions ingredients, there's lots of easier ways to get them than by destroying Dementors. The Aurors manage to get the things to put them on in the first place, after all.

Either Dumbledore or Quirrel took it, or it's long since locked away in the Department of Mysteries.

But yeah, as JoshuaZ points out, just as easily could be a Potion-of-Death. (Or heck, "potion-of-anthropomorphic-personification")

Or a potion of instant death if it instead stored the decay effect from the dementor.

Another thought: write down a description of a complex magical principle that you understand, but that the interdict of merlin would prevent someone else who was reading it from understanding. Use the parchment you wrote on as an ingredient in a potion to make a potion with the mental work needed to discover/comprehend that principle.

Poof, Interdict of Merlin loses its teeth entirely. :)

Another thought that occurred to me: Felix Felicis. No wonder it's hard to brew. Only way you could brew it is if you literally got lucky in the process of brewing it, by chance, so that you can take that "chance" and put it into the potion.

(hrm... might be able to automate the process of making Felix: Have a machine that keeps mixing the ingredients many times in parallel, ie, many "potential potions", and in the process does something like for each potential potion, has a coin (or some random bit source which can then be physically placed in the potion) which it flips a 100 times. It also tracks the results, and when one of the coins comes up all heads, it drops it into the candidate potion then calls up the wizard to complete the potion.)

Oh, and MoR!Snape did claim you could brew fame and stuff. That was one of the things MoR!Harry challenged him on, saying something like "How does that work anyways? You drink it and turn into a celebrity?"

Or, wait, an even more recursive version of your science power potion:

Make a clever potion. Use that potion as an ingredient in a potion to extract the mental work of creating a clever potion.

Use that potion as an ingredient... repeat. :)