What is east of Rhûn?

From the Encyclopedia of Arda:

Even Gandalf had never explored there, and though Aragorn had travelled there, we have no report of his doings.

Of its ancient geography we can glean a little from the Silmarillion:

• far beyond the Sea of Rhûn was another inland sea, the Sea of Helcar,

• and beyond that a range of red mountains known as the Orocarni.

• Somewhere in the lost east, too, lay Cuiviénen and Hildórien, where Elves and Men first awoke: all the Children of Ilúvatar could trace their ancestries back to the eastward regions of Middle-earth.

Sea of Helcar:

On its eastern shoreline lay the mountains of the Orocarni, and at their feet on a small bay was the land Cuiviénen, where the first Elves awoke in Middle-earth.

It was there that Oromë discovered them, and ultimately led many of them on a Great Journey around the northern end of the Sea of Helcar as they set out for Valinor in the distant west.

Orocarni:

A lost range of mountains in the far east of Middle-earth, near Cuiviénen, beneath which the first Elves awoke.

Called the Mountains of the East, the name Orocarni literally translates as 'Red Mountains'.

We know little more about them, except that they were apparently the source of the many streams that ran down into Cuiviénen.

Blue Wizards:

After landing in Middle-earth, they travelled directly into the far east with Curunír (better known as Saruman).

The mission of the Blue Wizards was to travel to eastern lands occupied by Sauron, and help to foment unrest among his subject peoples.

Whether they succeeded or not, even Tolkien was unsure, though he imagined that they had probably failed. It's also uncertain why they remained in the east while Curunír returned

There is not a great amount of detail about what is east of Rhun.

Taken from The lord of the rings wiki:

Of its ancient geography we can glean a little from The Silmarillion; far beyond the Sea of Rhûn was another inland sea, the Sea of Helcar, and beyond that a range of red mountains known as the Orocarni (Red Mountains). Somewhere in the lost east, too, lay Cuiviénen and Hildórien, where Elves and Men first awoke: all the Children of Ilúvatar could trace their ancestries back to the eastward regions of Middle-earth.

Some other facts:

Dorwinion is situated in Rhun and this is where the easterlings reside.

Four of the seven clans of the dwarves also live there.

It is also home to lost Elves such as Avari and Úmanyar.

There are no other countries or lands beyond Rhun. Like Russia it is all Rhun past the eastern borders of the map. To the southeast is another land called Khand.

I used to own (for a decade) this huge wall-map of all points in Middle Earth beyond Rhun, beyond the Northern Wastesand the Icebay of Forochel, and beyond Far Harad. I got it as a fold-out from this magazine I'd bought about Middle Earth, and I've seen thumbnail images of it online.

Beyond the great desert Haradwaith, and its dozen or so oasis points, there were heavily forested shores with one particular (rain)forest about twice the size as Mirkwood. And south of this, heavily forested islands. It even showed Beleriand, as it was before it sank, with a great depiction of that big forest in Beleriand ("Taur-En-something") compared to Mirkwood's size.

Out east of Rhun and Khand,ran a mountain-range notably grander than the Misty and Blue mountains.