Someday I'm gonna write an awesome, all-encompassing lesson about the te-form in Japanese.

This is not that lesson.

But it might be a piece of it: The difference between ～て and ～てて.

First, ～てて is the same thing as ～ていて. For example, 2.a. and 2.b., below, are identical:

1. 待って（まって // wait; hold on）
2.a. 待っていて（まっていて // wait; hold on）
2.b. 待ってて（まってて // wait; hold on）

Wait. Hold on! Why are all of these translated the same?

Well, I don't really know how to differentiate them in simple terms. Anyways, 2.a. and 2.b. are the exact same--待ってて is just a contraction of 待っていて. As you may have guessed, the contracted form is very common in spoken language.

But! 待って is not the same as the other two.

As is usually the case, this is best explained with an example...

待って means just "wait" in a very general sense. Usually it refers to waiting for a very short (and often nonexistent) period of time:

待って、今なんて言った？
まって、 いま　なんて　いった？
Wait, what did you just say?
Literally: "wait, + now + what + said?
Note: I translated なんて as "what," but I suppose a closer translation is something like "what (was said)."

↑ The speaker in this case is not really asking the listener to wait at all. They're just asking them to stop, to hold up for a second.

Compare that to the following usage of 待ってて：

トイレ行ってくるからここで待ってて。
トイレ　いって　くる　から　ここ　で　まってて。
I'm gonna go to the bathroom, so wait here.
Literally: "toilet + go (and) + come + because + here + で + wait."

The key, as you might have guessed, is this second て, which is a contraction of いて.

In other words, it is the auxiliary verb ～いる！And we use this auxiliary verb in order to mark that someone is (staying) in a certain state. That's why it is used for the present continuous tense also:

本読んでる。
ほん　よんでる。
I'm reading a book.
Literally: "book + am reading."

読んでる＝読んでいる

The ～いる is marking that this is an ongoing state.

So when we put that auxiliary verb onto the end of a te-form request, also in te-form, then the nuance is that we are asking the listener to remain in a certain state.

Understanding this conceptually might not be too difficult, but using it flawlessly can be tricky. Reading lessons like this will help, but ultimately you probably need to get lots of speaking practice (if you're interested in speaking, that is).

The following examples may also help...

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どっか行って。
どっか　いって。
Get out of my sight.
Literally: "somewhere + go."
Note: For instance, your significant other might say this to you if you manage to really piss them off.

↑ This te-form request is asking (well, telling) the listener to perform an action.

↓ But this double te-form request is asking the listener perform an action and remain in a state:

遅れそうだから先に行ってて。
おくれそう　だから　さき　に　いってて。
I'm gonna be late, so go ahead without me.
Literally: "looks like (I) will be late + (is) because + before + に + go (and stay)."

↑ For example, you could message this to a friend that is supposed to meet you at the station for a party somewhere. You'll be late, so you want them to go to the party before you. The emphasis is not on going. The emphasis is on going and being at the party. So we add a second て.

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眠かったら寝ていいよ。
ねむかったら　ねて　いい　よ。
If you're tired, (you can) go to sleep.
Literally: "if sleepy + sleep + good + よ."

↑ Encouraging an action.

↓ Encouraging a state.

寝てていいよ。
ねてて　いい　よ。
You don't have to get up.
Literally: "stay sleeping / stay lying down + good + よ."

For example, if you get home late and your significant other struggles to get up and greet you lovingly, you could say this to them.

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この絵を見てください。
この　え　を　みて　ください。
Please look at this picture [painting].
Literally: "this + picture / painting + を + look + please."

↑ Encouraging an action.

↓ Encouraging an ongoing action. In other words, encouraging a state!

見てて、もうすぐゴジラ出てくるから。
みてて、 もうすぐ　ゴジラ　でて　くる　から。
Keep watching. Godzilla's gonna show up soon.
Literally: "keep watching, + soon + Godzilla + come out (and) + come + because."

We're telling them to keep their eyes on the screen, so we add a second て.

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I still mess up this distinction from time to time, so I wouldn't stress about it too much. I mean, it's hard to think about this stuff when sentences are flying back and forth at the speed of conversation, yeah?

But knowing the difference will help you notice the distinction when used by native speakers. And eventually that will seep into your own Japanese, and you'll be the te-form master.

Well, closer to becoming the te-form master...