Asshai by the shadow is one the most interesting places in ASOIAF. Magical, mysterious and unexplored. Unfortunately, it will likely stay this way forever as GRRM has said that we won't see Asshai in the books.

-Will we see Asshai?

-Only in flashback and memory, if at all.

But i believe that it wasn't always suppose to be this way. And in ASOIAF's early days Asshai was going to play much bigger role. We were suppose to see it. But after abandoning 5 year gap while writing AFFC/ADWD, GRRM dropped this storyline.

Evidence for this:

• Asshai was mentioned 48 times total in the first 3 books, but only 18 times in the last two.
• In books 2 and 3, Quaithe almost explicitly says to Dany that she must go to Asshai.

“To go north, you must journey south. To reach the west, you must go east. To go forward you must go back, and to touch the light you must pass beneath the shadow.”

Asshai, Dany thought. She would have me go to Asshai. “Will the Asshai’i give me an army?” she demanded. “Will there be gold for me in Asshai? Will there be ships? What is there in Asshai that I will not find in Qarth?”

“Truth,” said the woman in the mask. And bowing, she faded back into the crowd.

• Asshai is tied to many important plot points and characters introduced in books 1-2. The dragon eggs, Mirri Maz Duur, Melisandre, the legend of Azor Ahai, Quaithe...

• Back in 2003, GRRM said that that there is remote possibility of us seeing Asshai in AFFC. link

As you can see, most of these points are related to Daenerys. And i am pretty sure that GRRM's original plan was to send Daenerys to Asshai after Slavers Bay.

Why did GRRM drop it

GRRM faced many problems while writing ADWD. Many of them revolved around so called Meereenese Knot. While trying to untie it, George changed Daenerys ADWD storyline drastically. For example, back in 2003, Drogon's return to Meeren happened in the very first Dany's spoiler chapter. In the book, it happened in her second to last chapter.

I think, Slaver's Bay storyline took much longer than George expected. And because of that, with GRRM still trying to end ASOIAF in 7 books, there is no time for Asshai.

What are are gonna get instead?

I don't think that GRRM complletely dropped this storyline. Whatever was going to happen in Asshai is probably very important, and is still happening in TWOW. Just not in Asshai, but in Volantis.

This city is barely mentioned in first 3 books. (10 times to be exact). But in ADWD, we see it from 3 different POV's. We learn about it's history through expositional lessons of Young Griff. But the biggest argument for it being out new Asshai is the red priests. Suddenly, Volantis is the center of R'hllor religion, and the priests are proclaiming Daenerys Azor Ahai. Wouldn't it make much more sense for it to happen in Asshai?

I think that originally, after Slavers Bay Dany was suppose to go to the dothraki, and then turn East and go to Asshai. But after realising that there is no time for this, George basically relocated this storyline, putting it on the way from Slaver's Bay to Westeros.

I also have a bit tinfoily suspicion that in pre-Volantis days Dany was never going to turn West. Instead she was going to travel only East, going past Asshai and eventually appearing in sustet sea, on the western shore of Westeros. Going east to travel west. Leading dothraki to world's end, as Stalion who Mounts the World in prophecised. I know, that this doesn't seem likely or even possible because of world's map. But remeber, back in the day, GRRM had pretty vague idea about the geography of his world. In fact, we know for sure that originally Essos was more compact and East of it was very different. So maybe originally Dany was meant to travel around te world. Maybe that's how Mance's Asshai silk ended up on Frozen Shore.

Also, this line seems like it could be a hint.

“The wine is ordinary. It is said that across the jade Sea they make a golden vintage so fine that one sip makes all other wines taste like vinegar. Let us take my pleasure barge and go in search of it, you and I.”

“The Arbor makes the best wine in the world,” Dany declared.

It does indeed. Arbor Gold.

Whether that theory is correct or not, it definitely isn't happening now. Daenerys's path lies West. To Volantis, Pentos and eventually - Westeros. And Asshai by The Shadow will forever stay a mystery.

TLDR: GRRM planned to send Dany to Asshai, but Meereense storyline took to long, so he replaced Asshai with Volantis.