What was the wish that Harry gave to Quirrell that was rejected?

Question

In Eli Yudkowsky's fanfiction Harry Potter and the Methods of Rationality, Professor Quirrell offers 'Quirrell points' to deserving students, to be used in a variety of ways:

"These and other after-school activities will also earn you Quirrell points. What are Quirrell points, you ask? The House point system does not suit my needs, because it makes House points too rare. I prefer to let my students know how they are doing more frequently than that. And on the rare occasions I offer you a written test, it will mark itself as you go along, and if you get too many related questions wrong, your test will show the names of students who got those questions right, and those students will be able to earn Quirrell points by helping you."

...wow. Why didn't the other professors use a system like that?

"What good are Quirrell points, you wonder? For a start, ten Quirrell points will be worth one House point. But they will earn you other favors as well. Would you like to take your exam at an unusual time? Is there a particular session you would very much prefer to skip? You will find that I can be very flexible on behalf of students who have accumulated enough Quirrell points. Quirrell points will control the generalship of the armies. And for Christmas - just before the Christmas break - I will grant someone a wish. Any school-related feat that lies within my power, my influence, or above all, my ingenuity. Yes, I was in Slytherin and I am offering to formulate a cunning plot on your behalf, if that is what it takes to accomplish your desire. This wish will go to whoever has earned the most Quirrell points within all seven years."

^{Emphasis added}

After a period of time, Harry - along with (warning - major spoiler)

Hermione and Malfoy

has his wish read by Professor Quirrell,

[..] Harry Potter visibly tensed up as the Defense Professor drew forth the parchment. “And Mr. Potter wishes for—”

There was a pause as Professor Quirrell looked at the parchment.

Then, without any change of expression on Professor Quirrell’s face, the sheet of parchment burst into flames, and burned with a brief, intense fire that left only drifting black dust sprinkling down from his hand.

“Please confine yourself to the possible, Mr. Potter,” said Professor Quirrell, sounding very dry indeed.

There was a long pause; Harry, standing beside Draco, looked rather shaken.

What in Merlin’s name did he ask for?

“I do hope,” said Professor Quirrell, “that you prepared another wish, if I could not grant that one.”

I can't say it any better: what in Merlin's name did he ask for?

I have read the work right through and the authors notes, but cannot recall encountering what Harry's wish was.

Answer 1

The answer can be found on page 602, when Harry is talking to Draco.

“So what did you wish the first time?” said Draco.
[...]
“It wasn’t really all that interesting,” Harry said with obviously artificial lightness. “Just, I wish Professor Quirrell would teach Battle Magic again next year.”

Quirrel refuses this wish because of the curse on the DADA teacher position, thus supposedly making it impossible for any teacher to last more than one year in the position.

While it would normally seem possible that Harry was lying about his wish, it's a plot point that Harry never outright lies about anything (except one thing, which happens later in the story). He can misdirect, be misleading, refuse to answer, but he doesn't outright lie until another 200+ pages in. The "obviously artificial lightness" is him faking that it's no big deal, when it was actually something extremely important to him.