Term for offering something just out of reach?

Look up tantalize. It is defined at Oxford dictionary as

Torment or tease (someone) with the sight or promise of something that is unobtainable.

Many writers have referred to...

(dangling) a mythical carrot

...but when I stop to think about it, I'm pretty sure the donkey - carrot - stick reference isn't to some kind of mythical fairy tale. I'm sure you really can get a donkey to traipse round a treadmill all day if you hang a real (non-mythical) carrot off a stick, just in front of him.

So that carrot can't be exactly "mythical" (besides which, if the realworld treadmill donkey never got a carrot, I'm sure he'd give up after a while; donkeys aren't that dumb). But it has been used.

Ask for the moon is an idiom meaning to ask for something that is unobtainable.

He may as well have asked me for the moon. I was never going to get that chair.

This is related to other moon idioms: Promise the moon: to offer something that's impossible.

Right now I'm so tired that I would promise the moon to be able to sit in that chair!

Cry at/for the moon: make an outlandish or unreasonable request, especially one that is unlikely to be fulfilled.

If he thinks he can get out of that chair, he's crying at the moon.

(Idioms from The Free Dictionary)