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Have me whole, I’m a friend to your legs
Remove one letter, I’m a friend to your head
Remove once more, I become ethereal
Remove one more time, I can’t be seen but felt

Who am I?

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You are a

chair

Because:

  1. A chair enables you to sit to give rest to your legs

  2. Remove c -> Hair keeps your head warm

  3. Remove h -> Air is quite ethereal

  4. Remove a -> IR (infrared) can't be seen but can be felt.

edit: Should've reloaded to see Stiv's same answer, although a slight difference in reasoning. Quite astonishing to see similar layout.

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  • 3
    $\begingroup$ I personally think your answer fits the fourth clue better than Stiv's $\endgroup$ – AHKieran Feb 10 at 16:04
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    $\begingroup$ @AHKieran Personally, I do too! Both last lines technically fit, but I imagine this might be the one the OP intended... $\endgroup$ – Stiv Feb 10 at 16:16
  • $\begingroup$ Sorry, @Stiv, but you are right :) Credits for a great answer though! $\endgroup$ – Ébe Isaac Feb 11 at 0:42
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    $\begingroup$ You can feel the fourth thing. $\endgroup$ – Ergwun Feb 11 at 2:09
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    $\begingroup$ Yeah "IR" is a really bad solution/hint IMO (assuming it was the authors intention). Infrared radiation is absorbed by the body, and heats you up. Most of the radiative heat you feel is in the IR spectrum AFAIK. $\endgroup$ – JMac Feb 11 at 15:50
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Perhaps you are a:

CHAIR

Have me whole, I’m a friend to your legs

A CHAIR is 'a friend to your legs' in that it enables them to rest when you sit on one...

Remove one letter, I’m a friend to your head

Remove the C --> HAIR - 'a friend to your head' in that hair keeps your head warm and protects it from the sun.

Remove once more, I become ethereal

Remove the H --> AIR - one of the meanings of 'ethereal' is 'light and airy'. What can be more so than air itself?!

Remove one more time, I can’t be felt or seen

Remove the R --> AI, or Artificial Intelligence - something which 'can't be felt or seen'.

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  • 1
    $\begingroup$ It could have been: "Have me whole, I’m a friend to your legs; Remove one letter, I’m a friend to your head; Remove once more, I'm a friend to your lungs, Remove one more time, I'm a friend to your computer" $\endgroup$ – Avi Feb 10 at 16:35
  • $\begingroup$ @Avi That's really neat! $\endgroup$ – Stiv Feb 10 at 16:49
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I know there's already an accepted answer, but this one also fits, to me:

Boots

because

High boots go up to your legs and protect from muck and cold

remove a letter to get

Boos, which is a brand name for cutting blocks, which in medieval times were associated with decapitation

one more letter becomes

Boo! An exclamation uttered by ghosts, traditionally, which are ethereal

And finally one last letter brings us to

BO, a frequent abbreviation for Body Odor; never felt or seen but only smelled!

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  • 1
    $\begingroup$ Oh yes decapitation, the greatest friend of heads all around! $\endgroup$ – theonlygusti Feb 11 at 20:45
  • $\begingroup$ @theonlygusti the chopping block knows the companionship of a great many heads! $\endgroup$ – Erin B Feb 11 at 21:45

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