What is the meaning of "praise someone's head off"?

Question

Your note just came and am utilizing a good hangover to answer it. I was not annoyed at anything you said (You surely know by now, I’ve written it often enough, how much I admire your work). I was only annoyed at your refusal to accept the sincere compliment G. Stein was making to you and instead try and twist it into a slighting remark. She was praising her head off about you to me when you came up she started to repeat it and then at the end of the praise to spare you blushes and not be rude to me she said that our flames (sic) were maybe not the same—then you brood on that—

This is the first paragraph of Ernest Hemingway's letter to F. Scott Fitzgerald. I looked up the dictionary for the phrase "head off" but still don't understand what does "praising her head off" mean in this context.

Answer 1

The idiom is listed under 'one's head off' in some dictionaries.

Merriam-Webster

one's head off (idiom)

: very loudly or without restraint

She had to shout her head off to get her friends attention at the concert.

He is shy at first, but will talk your head off [=will talk continuously] once you get to know him.

You can yell your head off at him, but he still won't listen.

They will be laughing their heads off when they see my costume.

OED:

to —— one's head off: to —— with great vigour or intensity, at great length, etc., esp. to the point of weariness or exhaustion.

1890 C. C. Harrison Anglomaniacs ii. 79 What man wants to work his head off to lay up money, and then see a fool and profligate walk away with it?

1951 J. Cornish Provincials 11 As term progressed, Saturdays and Sundays..we would sit in our den..talking our heads off.

1985 J. Howker Nature of Beast i. 21 We spent this pretty miserable day..not talking much, just smoking our heads off.

2010 C. Grant Teenie ix. 91 Those girls look like straight-up groupies, screaming their heads off and trying to touch Greg.